Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Other approaches to simplify $\frac{\tan^2x-\sin^2x}{\tan 2x-2\tan x}$ I want to simplify the trigonometric expression $\frac{\tan^2x-\sin^2x}{\tan 2x-2\tan x}$.
My approach,
Here I used the abbreviation $s,c,t$ for $\sin x$ and $\cos x$ and $\tan x$ respectively,
Numerator is, $$\frac{s^2}{c^2}-s^2=\frac{s^2-s^2c^2}{c... | My method uses Euler's formula and Computer Algebra Systems.
Let $\,X := e^{i\,x}.\,$ Then
$$ s := \sin(x) = \frac{X-X^{-1}}{2i}, \;\;
t := \tan(x) = \frac{X-X^{-1}}{i(X+X^{-1})}, \\
c := \cos(x) = \frac{X+X^{-1}}{2}, \;\;
t_2 := \tan(2x) = \frac{X^2-X^{-2}}{i(X^2+X^{-2})}. $$
Simplify a little to get
$$ s = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4510735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Which number is bigger: $e^{\sqrt{5}} + e^{\sqrt{7}}$ or $2e^{\sqrt{6}}$? Let $f:(0, + \infty) \rightarrow \mathbb{R}$ be $f(x) = e^{\sqrt{x}}$.
Which number is bigger: $e^{\sqrt{5}} + e^{\sqrt{7}}$ or $2e^{\sqrt{6}}$?
I would like to ask for a hint on how to approach this question, I don't have an idea.
| We have
\begin{align*}
e^{\sqrt{5}} > 2^{\sqrt{5}} > 2^1 \implies \frac{1}{e^\sqrt{5}} < \frac{1}{2} \tag{$*$}\\
e^{\sqrt{7}} > 2^{\sqrt{7}} > 2^1 \implies \frac{1}{e^\sqrt{7}} < \frac{1}{2} \tag{$**$}
\end{align*}
Now because $\ln{x}$ is an increasing function, we can write:
\begin{align*}
\ln(e^{\sqrt{5}} + e^{\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4510895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If $x^2+2(\alpha-1)x-\alpha+7=0$ has distinct negative solutions... Let $\displaystyle{ \alpha }$ be real such that the equation $\displaystyle{ x^2+2(\alpha-1)x-\alpha+7=0 }$ has two different real negative solutions. Then
*
*$ \ \displaystyle{ \alpha<-2 }$ ;
*$ \ \displaystyle{ 3<\alpha<7 }$ ;
*it is impossible... | Sum of roots$=-2(\alpha-1)<0\implies \alpha>1$
Product of roots$=(7-\alpha)>0\implies\alpha<7$
with the condition that the discriminant, $(2\alpha-2)^2-4(7-\alpha)>0\\\implies (\alpha-3)(\alpha+2)>0\\\implies \alpha>3\ or\ \alpha<-2\\
\therefore 3<\alpha<7$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Prove if $\sqrt{x+1}+\sqrt{y+2}+\sqrt{z+3}=\sqrt{y+1}+\sqrt{z+2}+\sqrt{x+3}=\sqrt{z+1}+\sqrt{x+2}+\sqrt{y+3}$, then $x=y=z$. Let $x$, $y$, $z$ be real numbers satisfying $$
\begin{align}
&\sqrt{x+1}+\sqrt{y+2}+\sqrt{z+3}\\
=&\sqrt{y+1}+\sqrt{z+2}+\sqrt{x+3}\\
=&\sqrt{z+1}+\sqrt{x+2}+\sqrt{y+3}.
\end{align}$$
Prove that... | Not sure if this is a good approach, but:
Let $f(t) = \sqrt{t+2}-\sqrt{t+1}$. Observe $f$ is strictly decreasing. Then subtract $\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}$ from each part of the equation to get: $$\begin{align}
&f(y)+f(z)+f(z+1)\\
=&f(z)+f(x)+f(x+1)\\
=&f(x)+f(y)+f(y+1)
\end{align}$$
This reduces to two variable... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 1
} |
Why this transformation matrix $A$ has $\begin{pmatrix}0 \\ 1\end{pmatrix}$ as Eigenvector? I have the following transformation matrix:
$$
A=\begin{pmatrix}
1 & 0 \\
-1 & 4
\end{pmatrix}
$$
If I resolve to find the eigenvalues I get:
$$
\begin{vmatrix}
A-\lambda I
\end{vmatrix} = 0
$$
which leads to:
$$
\lambda_1 = 1;... | Straightforward calculation shows that:
$$A\begin{pmatrix}0 \\ 1 \end{pmatrix}=\begin{pmatrix}0 \\ 4 \end{pmatrix}=4\begin{pmatrix}0 \\ 1 \end{pmatrix}$$
So $\begin{pmatrix}0 \\ 1 \end{pmatrix}$ is an eigenvector corresponding to the eigenvalue $\lambda=4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$x_2 + x_3 + x_4 + x_5 = n$ where $x_i$ is not divisible by $i$? How can I find the generating function for the number of solutions of non-negative integers for the equation $x_2$ + $x_3$ + $x_4$ + $x_5 = n$ where $x_i$ is not divisible by $i$?
Attempt:
I know $x_2$ is a variable for odd numbers and its generating func... | You have to multiply the series
$\left(\sum_{i \geq 0 \\ \text{is not} \\ \text{divisible} \\ \text{by 2} } x^i \right)$$\left(\sum_{i \geq 0 \\ \text{is not} \\ \text{divisible} \\ \text{by 3} } x^i \right)$$\left(\sum_{i \geq 0 \\ \text{is not} \\ \text{divisible} \\ \text{by 4} } x^i \right)$$\left(\sum_{i \geq 0 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4511941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\frac {x \csc x + y \csc y}{2} < \sec \frac {x + y}{2}$
If $0 < x,y < \frac {\pi}{2}$, prove that:
$$
\frac {x \csc x + y \csc y}{2} < \sec \frac {x + y}{2}
$$
My attempt. First, I tried to change this inequality:
$$
\frac {\frac{x}{\sin x} + \frac{y}{\sin y}}{2} < \frac{1}{\cos \frac{x+y}{2}}
$$
Then,it'... | We have
$$\frac{x}{\sin x} = \frac{x}{2\sin \frac{x}{2} \cos \frac{x}{2}} = \frac{x}{4\sin \frac{x}{4}\cos\frac{x}{4} \cos \frac{x}{2}} = \frac{\frac{x}{4}}{\tan \frac{x}{4} \cos^2\frac{x}{4}\cos\frac{x}{2}} \le \frac{1}{\cos^2\frac{x}{4}\cos\frac{x}{2}}$$
where we have used $\frac{x}{4} \le \tan \frac{x}{4}$.
Also, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4512437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solve $(x^3+1)=2\sqrt[3]{2x-1}$ algebraically? I'm trying to solve the said equation in the thread title algebraically.
$$(x^3+1)=2\sqrt[3]{2x-1}$$
Cubing both sides and simplifying:
$$x^9+3x^6+3x^3-16x+9 = 0$$
Not sure if this can be solved algebraically?
Edit: WA gives $3$ solutions $x=1,\frac{1}{2}(-1-\sqrt{5}),\fra... |
$$f(x)=x^9+3x^6+3x^3-16x+9$$
$$f(x)=0\Rightarrow (x-1)(x^2+x-1)(x^6+2x^4+2x^3+4x^2+2x+9)=0$$
where $$x^6+2x^4+2x^3+4x^2+2x+9=x^6+x^4+x^2(x+1)^2+2x^2+(x+1)^2+8>0$$ for $\forall x\in \mathbb{R}$, so we get:
$$(x-1)(x^2+x-1)=0$$
There are three real roots:
$$x_1=1, x_2=\frac{-1-\sqrt{5}}2, x_3=\frac{-1+\sqrt{5}}2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4513552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find projection point on elipse Given the ellipse
$$
E(x, \ y) = ax^2 + bxy + cy^2 + dx + ey + f = 0
$$
With $4ac - b^2 = 1$
How can I compute the projection $(x_p, \ y_p)$ of the point $(x_0, y_0)$ in this ellipse?
$$
E(x_p, \ y_p) = 0
$$
Motivation: There is already a direct algorithm to fit ellipses from a set of da... | This answer is not the one that I expected but it solves my problem in a non-elegant way.
I will divide this answer in three parts:
*
*Transform the ellipse with center $(x_c, \ y_c)$ and rotated counter clockwise with angle $\varphi$
$$E(x, y) = ax^2+bxy+cy^2 + dx + ey+f = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
into anot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4513839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Bounding a simple ratio of bivariate functions For any $x,y \ge 0$, define $f(x,y) := x^2+y^2 + 2c xy$, where $c := \sqrt{2/\pi}$. It is clear that
$$
\sqrt{2/\pi}\cdot (x+y)^2 \le f(x,y) \le (x+y)^2.
\tag{1}
$$
This is an immediate consequence of the fact that $c \le 1 \le 1/c$.
Question. What is a good upper-bound fo... | Since $c<1$ and $f(x,y) =(x+y)^2-2xy(1-c)$ we have
$$\frac1\alpha := \inf_{x,y}\dfrac{f(x,y)}{(x+y)^2}= 1-(1-c)\sup_{x,y}\frac{2xy}{(x+y)^2}$$
It remains to compute
$$\sup_{x,y}\frac{2xy}{(x+y)^2}= \sup_{x,y}\frac12\frac{(x+y)^2-(x-y)^2}{(x+y)^2}= \frac12-\frac12\inf_{x,y}\frac{(x-y)^2}{(x+y)^2}=\frac12$$
Indeed since ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4516621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is $f(x)=\frac{x}{e^x-1}$ continuous? My exercise says:
$$f(x)=\frac{x}{e^x-1},\quad x\neq0$$
$$f(0)=1$$
Can someone explain whatever this means? And why does the graph not cut off at $x=0$?
Edit : Apologies ,it's not $f(0)=0$ but $f(0)=1$.
And to clarify the exercise asks to confirm that $f'(0)$ exists (But that's... | $\frac{x}{e^x- 1}$ is NOT continuous. In order for a function, f(x), to be continuous at x= a, three things must be true.
*
*$\lim_{x\to a} f(x)$ exist.
*f(a) exist.
*$\lim_{x\to a} f(x)= f(a)$.
(For 3 to make sense, 1 and 2 must be true so often only
3 is stated.)
$\lim_{x\to 0} \frac{x}{e^x- 1}$ exists and is 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4518652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solving $\frac{dx}{dt}=\frac{xt}{x^2+t^2},\ x(0)=1$ I have started self-studying differential equations and I have come across the following initial value problem
$$\frac{dx}{dt}=\frac{xt}{x^2+t^2}, \quad x(0)=1$$
Now, since $f(t,x)=\frac{xt}{x^2+t^2}$ is such that $f(rt,rx)=f(t,x)$ for every $r\in\mathbb{R}\setminus\{... | $$\frac{dx}{dt}=\frac{xt}{x^2+t^2},\ x(0)=1$$
Multiply by $2x$:
$$2x{dx}=\frac{2x^2t}{x^2+t^2}dt$$
$${dx^2}=\frac{x^2}{x^2+t^2}dt^2$$
Substitute $u=x^2,v=t^2$:
$${du}=\frac{udv}{u+v}$$
This is a first order linear DE $uv'=u+v$ that you can also solv as this:
$$u{du}={udv}-vdu$$
$$\dfrac {du}u=\dfrac {udv-vdu}{u^2}$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4520385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Given $3x+4y=15$, $\min(\sqrt{x^2+y^2})=?$ (looking for other approaches)
Given, $(x,y)$ follow $3x+4y=15$. Minimize $\sqrt{x^2+y^2}$.
I solved this problem as follows,
We have $y=\dfrac{15-3x}{4}$,
$$\sqrt{x^2+y^2}=\sqrt{x^2+\frac{(3x-15)^2}{16}}=\frac{\sqrt{25x^2-90x+225}}4=\frac{\sqrt{(5x-9)^2+144}}{4}$$Hence $\mi... | By Cauchy-Schwarz inequality,
$\sqrt{x^2+y^2}\ge\frac{15}{\sqrt{3^2+4^2}}=3$
and by the case of Cauchy-Schwarz equality, this value is attained, hence it is the min you look for.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4525324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Show that $\frac{1-\sin2\alpha}{1+\sin2\alpha}=\tan^2\left(\frac{3\pi}{4}+\alpha\right)$ Show that $$\dfrac{1-\sin2\alpha}{1+\sin2\alpha}=\tan^2\left(\dfrac{3\pi}{4}+\alpha\right)$$
I am really confused about that $\dfrac{3\pi}{4}$ in the RHS (where it comes from and how it relates to the LHS). For the LHS:
$$\dfrac{1... | $$\dfrac{1-\sin2\alpha}{1+\sin2\alpha} = \left(\frac{\tan \alpha-1}{\tan \alpha+1}\right)^2= \left(\frac{1-\tan \alpha}{1+\tan \alpha}\right)^2$$
$$=\tan^2\left(\dfrac{\pi}{4}-\alpha\right) =\tan^2\left(\alpha-\dfrac{\pi}{4}\right)$$
Inverse tangent function satisfies two angles in range $0, 2 \pi.\;$We are allowed to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4526177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 2
} |
Find the range of $f(x)=\frac1{1-2\sin x}$ Question:
Find the range for $f(x)= 1/(1-2\sin x)$
Answer :
$ 1-2\sin x \ne 0 $
$ \sin x \ne 1/2 $
My approach:
For range :
$ -1 ≤ \sin x ≤ 1 $
$ -1 ≤ \sin x < 1/2$ and $1/2<\sin x≤1 $ , because $\sin x≠1/2$
$ -2 ≤ 2\sin x <1 $ and $ 1< 2\sin x≤2 $
$ -1 < -2\sin... | You made an error when you took reciprocals. When $a, b$ have the same sign, $$a < b \implies \frac{1}{a} > \frac{1}{b}$$
You failed to reverse the direction of the inequalities when you took reciprocals.
Since the denominator cannot be zero, you correctly concluded that the function is defined for those real numbers ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4526772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Find all values of a so that the circle $x^2 - ax + y^2 + 2y = a$ has the radius 2 My goal is to find all values of "a" so that the circle $x^2 - ax + y^2 + 2y = a$ has the radius 2
The correct answer is: $a = -6$ and $a = 2$
I tried solving it by doing this:
$x^2 - ax + y^2 +2y=a$
$x^2 - ax + (y+1)^2-1=a$
$(x - \frac ... | $\frac{a^2+4a+4}{4}$ should equal $2^2=4$, not $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4527455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the all possible values of $a$, such that $4x^2-2ax+a^2-5a+4>0$ holds $\forall x\in (0,2)$
Problem: Find the all possible values of $a$, such that
$$4x^2-2ax+a^2-5a+4>0$$
holds $\forall x\in (0,2)$.
My work:
First, I rewrote the given inequality as follows:
$$
\begin{aligned}f(x)&=\left(2x-\frac a2\right)^2+\fr... | Might be calculative but this works
Find the roots using quadratic formula then if the condition satisfies then there are 3 cases
$$\text{Case1}:-$$ $$\text{first the smaller of the 2 roots (the one with the negative sign) is greater than 2 this gives the quadratic}$$ $a^2-13a-68\ge0$ and $3a^2-20a+16\le0$
$$\text{Case... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4528746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 11,
"answer_id": 7
} |
How to calculate the gradient of $\mathbf{x}^T\mathbf{W}^2(\mathbf{W}^2)^T\mathbf{x}$ w.r.t. $\mathbf{W}$? I need to calculate the gradient of $\mathbf{x}^T\mathbf{W}^2(\mathbf{W}^2)^T\mathbf{x}$ w.r.t. $\mathbf{W}$. Here is what I have tried. Let $A=W^2$, then the form reduces to
\begin{align*}
\frac{\partial \mathbf{... | Let
$\mathbf{y}=(\mathbf{W}^2)^T \mathbf{x}$.
It holds
\begin{eqnarray*}
d\phi
&=& 2 \mathbf{y}:d\mathbf{y} \\
&=& 2 \mathbf{y}\mathbf{x}^T:d(\mathbf{W}^2)^T \\
&=& 2 \mathbf{B}:d(\mathbf{W}^2) \\
&=& 2
\left[\mathbf{W}^T\mathbf{B}+
\mathbf{B} \mathbf{W}^T \right]
:d\mathbf{W}
\end{eqnarray*}
where
$\mathbf{B}
=\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4529245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $\left|\int_a^bf(x)dx \right|\le \frac{(b-a)^3}{24} \max_{a\le x\le b}|f''(x)|$ Suppose that $f\in C^2 [a, b]$ and that $f(\frac{a+b}{2}) =0$ then prove that \begin{equation} \bigg|\int_a^bf(x)dx \bigg|\le \frac{(b-a)^3}{24} \max_{a\le x\le b}|f''(x)| \end{equation}.
I know that since $f\in C^2$, by the Taylor e... | Solution:
The Taylor expansion (with Lagrange remainder) of $f(x)$ at the point of $x=\dfrac{a+b}{2}$ is
\begin{align}
f(x)&=\color{blue}{\underbrace{f\left(\frac{a+b}{2}\right)}_{0}}+f^\prime\left(\frac{a+b}{2}\right)\left(x-\frac{a+b}{2}\right)+\frac{1}{2}f^{\prime\prime}\left(\xi\right)\left(x-\frac{a+b}{2}\right)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4535917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Solve the equation $\frac{\sqrt[7]{x-\sqrt2}}{2}-\frac{\sqrt[7]{x-\sqrt2}}{x^2}=\frac{x}{2}\sqrt[7]{\frac{x^3}{x+\sqrt2}}$ Solve the equation $$\dfrac{\sqrt[7]{x-\sqrt2}}{2}-\dfrac{\sqrt[7]{x-\sqrt2}}{x^2}=\dfrac{x}{2}\sqrt[7]{\dfrac{x^3}{x+\sqrt2}}$$ We have $x\ne0;-\sqrt2$.
Let's multiply both sides of the equation b... | Starting where you left off, at $(x^2-2)^8=x^{24}$, rearrange the equation to:
$$x^{24} - (x^2-2)^8 = 0$$
You could use the Binomial Theorem to expand the $(x^2-2)^8$ term, but for now I won't. Just note that the polynomial's constant term (and the product of all its roots) is $-(-2)^8 = -256$.
If you just want real s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4536302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
Find $\cos\frac{\pi}{12}$ given $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} -1}{2 \sqrt{2}}$
From a question I asked before this, I have trouble actually with the numbers manipulating part.
Using trigo identity, $\sin^2 \frac{\pi}{12} ... | Nothing wrong.
If you prefer, we can do some simplification.
Let me focus on $\sqrt{2+\sqrt3}$.
Let $$\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3}=x$$
$$2+\sqrt3+2-\sqrt3+2=x^2$$
Hence, we have $$\sqrt{2+\sqrt3}+\sqrt{2-\sqrt3}=\sqrt6.$$
Similarly, we have $$\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$$
Hence $$\sqrt{2+\sqrt3}=\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4539557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Arithmetic-Geometric limit $\lim\limits_{n\to\infty} x_n = \lim\limits_{n\to\infty} y_n$
For $x,y>0$, define two sequences $(x_n)$ and $(y_n)$ by $x_1=x,y_1=y$ and $x_{n+1}=(x_n+y_n)/2$ and $y_{n+1}=\sqrt{x_ny_n}$. Prove that $\lim\limits_{n\to\infty} x_n = \lim\limits_{n\to\infty} y_n= \dfrac{\pi}{\int_0^\pi \dfrac{d... | I will handle the integral at the end using the transformation given by Gauss. An alternative transformation is available on my blog (linked in comments to question).
Let us write $$I(x, y) =\int_0^{\pi/2}\frac{dt}{\sqrt{x^2\cos^2t+y^2\sin^2t}}\tag{1}$$ for $x, y>0$ and we prove $$I(x, y) =I\left(\frac{x+y} {2},\sqrt {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4543087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Show that discretization matrix is positive-definite
Given the following coefficient matrix $A^h$, resulting from the finite difference approximation of the biharmonic equation on a mesh with mesh size $h$:
\begin{equation*}
A^h = \frac{1}{h^4}
\begin{pmatrix}
5 & -4 & 1 & & \\
-4 & 6 & \ddots & \ddots... | Every symmetric, positive-definite matrix has a square root.
In particular, the root may be asked to be symmetric and positive-definite as well, and then it is uniquely determined.
For the given discretisation matrix $A^h$, which is highly structured, (t-)his square root
$$\sqrt{A^h} \;=\; \frac1{h^2}
\begin{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4546583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
easy way to calculate the limit $\lim_{x \to 0 } \frac{\cos{x}- (\cos{x})^{\cos{x}}}{1-\cos{x}+\log{\cos{x}}}$ I have been trying to use L'Hôpital over this, but its getting too long, is there a short and elegant solution for this?
The Limit approaches 2 according to wolfram.
| Start with $$\cos(x)=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6) \\
\log(\cos(x))=-\frac{x^2}{2}-\frac{x^4}{12}+O(x^6) \\
\cos(x)\log(\cos(x))=-\frac{x^2}{2}+\frac{x^4}{6}+O(x^6)$$
Then $$\cos(x)-e^{\cos(x)\log(\cos(x))}=1-\frac{x^2}{2}+\frac{x^4}{24}+O(x^6)-\left\{1+\cos(x)\log(\cos(x))+\frac{(\cos(x)\log(\cos(x)))^2}{2}+O\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4547927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
The convergence speed of $ \int_0^{\frac{\pi}{2}} \sin ^n(x) \operatorname{d}x $? I have already known how to prove
\begin{equation*}
\lim _{n \rightarrow \infty} \int_0^{\frac{\pi}{2}} \sin ^n(x) \operatorname{d}x = \sqrt{\frac{\pi}{2n}}
\end{equation*}
with Wallis's formula
\begin{equation*}
\quad \frac{... |
Are there any more powerful tools, like numerical methods to calculate the intergration?
Yes. There are.
Interval Integration
e.g.
We can take the sine to the n power of x intervals and then integrate those intervals.
This would probably be the easiest way to solve it:
$$
\begin{align*}
z &= \int_{0}^{\frac{\pi}{2}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4548070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Computing the eigenvalues and eigenvectors of a $ 3 \times 3$ with a trick The matrix is: $
\begin{pmatrix}
1 & 2 & 3\\
1 & 2 & 3\\
1 & 2 & 3
\end{pmatrix}
$
The solution says that
$ B\cdot
\begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix}
=
\begin{pmatrix}
6 \\
6 \\
6\end{pmatrix}$
$ B\cdot
\begin{pmatrix}
1 \\
1 \\
-1
... | There are many ways of finding eigenvectors and eigenvalues, but they all come down to solving a linear system of equations. For example, in this case you can use the fact that $\lambda_1=6$ is an eigenvalue (which you can easily find by solving the characteristic equation $-\lambda^3 + 6\lambda^2 - 9\lambda = 0$), to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4551846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Spivak, Ch. 22, "Infinite Sequences", Problem 1(iii): How do we show $\lim\limits_{n\to \infty} \left [\sqrt[8]{n^2+1}-\sqrt[4]{n+1}\right ]=0$? The following is a problem from Chapter 22 "Infinite Sequences" from Spivak's Calculus
*
*Verify the following limits
(iii) $\lim\limits_{n\to \infty} \left
[\sqrt[8]{n^2... | Consider a different approach as seen by the following.
\begin{align}
\sqrt[m]{n^p + 1} - \sqrt[m]{n^p} &= \sqrt[m]{n^p} \, \left( \sqrt[m]{1 + \frac{1}{n^p}} - 1 \right) \\
&= \sqrt[m]{n^p} \, \left( e^{\frac{1}{m} \, \ln\left(1 + \frac{1}{n^p}\right)} - 1 \right) \\
&= \sqrt[m]{n^p} \, \left( \frac{1}{m} \, \ln\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4552533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic is $9x^2+20x+10$
It is z... | Expanding on Anne Bauval's comment.
Rearranged equation is $(x^2+x+1)(3x+7)+2=0$
Discriminant of $x^2+x+1$ is negative. Thus, this quadratic is always positive.
$-2\lt x\lt4\implies-6\lt3x\lt12\implies1\lt3x+7\lt19$
It means the rearranged equation, on the given interval is (positive)(positive)+positive. Thus, never ze... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4553039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Closed Form Formula for Nonlinear Recurrence $a_{n+1}=\frac{a_{n}}{2} + \frac{5}{a_{n}}$ I'm trying to find a closed form solution to the sequence
$a_{n+1}=\frac{a_{n}}{2} + \frac{5}{a_{n}}$
I tried using a generating function approach in the following way:
Let $$f(x) = \sum_{n=1}^\infty a_n x^n$$
Then multiplying the ... | Let's make a change of variable $x_n = \frac{a_n}{\sqrt{10}}$, then
$$x_{n+1} = \frac{x_n^2 +1}{2x_n}$$
We have
$$x_{n+1} -1 = \frac{(x_n -1)^2}{2x_n}$$
$$x_{n+1} + 1 = \frac{(x_n +1)^2}{2x_n}$$
Then
$$\frac{x_{n+1} -1}{x_{n+1} + 1} = \left(\frac{x_n -1}{x_n +1}\right)^2$$
By reccurence, we obtain
$$\frac{x_{n} -1}{x_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4553375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Given any $a,b,c \geq 1$, prove that $a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$ Given any $a,b,c \geq 1$, prove that:
$a^2 + b^2 + c^2 \geq 2a\sqrt{b-1} + 2b\sqrt{c-1} + 2c\sqrt{a-1}$
I tried using most of the popular inequalities and I didn't end up anywhere. Can anyone guide me through this pro... | $RHS^2 \leq (4a^2+4b^2+4c^2)(a+b+c-3)$, so it suffices to show $a+b+c -3 \leq \frac{1}{4}(a^2+b^2+c^2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4554186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I prove this limit result associating with an infinite nested radical Let $a_n=\sqrt{1+\sqrt{2+\sqrt{3+...+\sqrt{n}}}}$
I can show that $\lim\limits_{n\to∞}a_n$ converges,let $l=\lim\limits_{n\to∞}a_n$
Now what puzzles me is that how to prove $\lim\limits_{n\to∞}\sqrt{n}\sqrt[n]{l-a_n}=\frac{\sqrt{e}}{2}$
I hav... | This is more like long observation with I think it fits better in this section than in the comment section.
It seems that your bounding came a little too soon (as comedians would say).
For each $n$ and $1\leq k\leq n$ set
\begin{align}
x_n&:=\sqrt{1+\sqrt{2+\ldots +\sqrt{n}}}\\
x_{k,n}&:=\sqrt{k+\sqrt{(k+1)+\ldots +\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4556694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding the big-$O$ estimate of the solutions to $\cot x = x$ So a problem for my class I am asked to find the big-$O$ estimate (up to 3 terms) of the solutions to $\cot x = x$. We label the solutions to this equation in increasing order $0<x_1<x_2<\cdots$ and we wish to find the expansion of $x_n$ as $n \rightarrow\in... | You already find the first term $n\pi$ for $x_n$ (with $0=x_0<x_1<...$). So, let's denote $x_n=n\pi+y_n$ with $y_n=\mathcal{o}(n)$ and $0<y_n<\frac{\pi}{2}$.
$$\begin{align}
&\Longrightarrow \cot(y_n)=n\pi+y_n \xrightarrow{n\to+\infty}+\infty \\
&\Longrightarrow y_n \xrightarrow{n\to+\infty} 0
\end{align}$$
So $y_n=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4558425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Determine the values of c and d so that E[Y] = c and Var(Y ) = 1 Question
Supppose $X$ is a random PDF $f(x)=\frac{2x}{c^{2}d^{2}}, \quad 0<x<cd$ where $c > 0, \; d > 0$.
Determine the values of $c$ and $d$ so that $\mathbb{E}[X]= c $ and $\mathrm{Var}(x) = 1$
My approach
My approach has been to try and isolate the var... | Here's a quick way to solve this. Brief hints are provided for the path to follow. Reveal the spoilers if you get stuck in computations.
You have $\mathbb E[X]=c$. Then put this into the Variance equation.
\begin{align*}1=\mathrm Var[X]&=\mathbb E[X^2]-\mathbb E[X]^2\\&=\int_{0}^{cd}x^2\frac{2x}{c^2d^2}\ \mathrm dx-c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4560936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How can we generalize factorisation of $(a+b)^n-(a^n+b^n)$
How can we generalize factorisation of
$$(a+b)^n-(a^n+b^n)\,?$$
where $n$ is an odd positive integer.
I found the following cases:
$$(a+b)^3-a^3-b^3=3ab(a+b)$$
$$(a+b)^5-a^5-b^5=5 a b (a + b) (a^2 + a b + b^2)$$
$$(a+b)^7-a^7-b^7=7ab(a+b)(a^2+ab+b^2)^2$$
Exap... | For $n=9$ we have
$$
(a+b)^9-(a^9+b^9)=3(3a^6 + 9a^5b + 19a^4b^2 + 23a^3b^3 + 19a^2b^4 + 9ab^5 + 3b^6)(a + b)ab,
$$
where the first part is irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4563475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Solve the equation $\sqrt{x^2+x+1}+\sqrt{x^2+\frac{3x}{4}}=\sqrt{4x^2+3x}$ Solve the equation $$\sqrt{x^2+x+1}+\sqrt{x^2+\dfrac{3x}{4}}=\sqrt{4x^2+3x}$$
The domain is $$x^2+\dfrac{3x}{4}\ge0,4x^2+3x\ge0$$ as $x^2+x+1>0$ for every $x$. Let's raise both sides to the power of 2: $$x^2+x+1+x^2+\dfrac{3x}{4}+2\sqrt{(x^2+x+1... | Let me try. One can rewrite equation as below.
$$\sqrt{x^2+x+1} + \frac{1}{2}\sqrt{4x^2+3x} = \sqrt{4x^2+3x},$$
$$\sqrt{x^2+x+1} = \frac{1}{2}\sqrt{4x^2+3x},$$
$$x^2+x+1 = \frac{1}{4}(4x^2+3x)$$
$$4x^2+4x+4 = 4x^2+3x,$$
$$x = -4.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4566074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solve a first order PDE derived from exponential generating function The PDE comes from a counting problem:
Suppose there is a bag of $m$ red, $n$ blue balls, and each time one randomly removes one ball until there are only balls of one color in the bag. What is the expected number of balls left?
Let $f(m,n)$ be the ex... | You have
$$x F_{x} + y F_{y} = (x + y) F + x e^{x} + y e^{y}$$
Using the method of characteristics in the parameterisation invariant form, we have
$$\frac{dx}{x} = \frac{dy}{y} = \frac{dF}{(x + y) F + x e^{x} + y e^{y}}$$
Solving across the first equality gives
$$\frac{x}{y} = C_{1}$$
Using this result, we can rewrite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4566287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing vector to equal vector forces/magnitudes I have this problem with its respective solution:
Assuming that the superior vector is $\overrightarrow{A}$, the middle vector is $\overrightarrow{B}$, and the inferior vector is $\overrightarrow{F}$
My computation was:
$$\overrightarrow{A} \cdot \overrightarrow{B}=\o... | Although I don't recommend it for this application, one can get a vector solution using $$\vec{A}\times \vec{B}=S\vec{F}\times \vec{B}$$
Where variable $S$ is the unknown force.
$$
\vec{A} \, = \, \left( \begin{align}20000 \; \cos \left( 2 \cdot \frac{\pi }{9} \right) \\ 20000 \; \sin \left( 2 \cdot \frac{\pi }{9} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4568190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Stuck on Liebniz Integration Technique This is a problem from the textbook Advanced Calculus by Edwin Wilson from Chapter 11 (page 281-288). The problem asks,
Evaluate by any means the following: $$\int_0^{\frac{\pi}{2}} \frac{ln(1 + \cos{a}\cos{x})}{\cos{x}} \, dx$$
I think I'm on the right track... but I'm getting st... | Let $\alpha := \cos(a)$ and $\beta := \sqrt{1-\alpha}$ and $\gamma := \sqrt{1+\alpha}$. (The latter two are well defined since $\cos(a) \in [-1,1]$.) Then the integral of concern is, just with some factorizations,
$$\mathcal{I}:= \int_0^1 \frac{1}{1+t^2 +\alpha - \alpha t^2} \, \mathrm{d}t
= \int_0^1 \frac{1}{t^2 \beta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4570259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
I can't find the solution of $\lim_{x\rightarrow 0} \left(1+\frac{x}{(x-1)^2}\right)^{\frac{1}{\sqrt{1+x}-1}}$ I can't find the solution of
$$\lim_{x\rightarrow 0} \left(1+\frac{x}{(x-1)^2}\right)^{\frac{1}{\sqrt{1+x}-1}}$$
Computing for $x$ goes to $0$ it gives a $1^\infty$ type of indeterminate form. I tried to solve... | Given
$$\lim_{x\rightarrow 0} \left(1+\frac{x}{(x-1)^2}\right)^{\frac{1}{\sqrt{1+x}-1}}$$ we have with $u= e ^{ \ln u}$,
$$ \lim_{x \to 0} \left(\frac{x}{\left(x - 1\right)^{2}} + 1\right)^{\frac{1}{\sqrt{x + 1} - 1}} = \lim_{x \to 0} e^{\ln{\left(\left(\frac{x}{\left(x - 1\right)^{2}} + 1\right)^{\frac{1}{\sqrt{x + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4571961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Seeking for other methods to evaluate $\int_0^{\infty} \frac{\ln \left(x^n+1\right)}{x^n+1} dx$ for $n\geq 2$. Inspired by my post, I go further to investigate the general integral and find a formula for
$$
I_n=\int_0^{\infty} \frac{\ln \left(x^n+1\right)}{x^n+1} dx =-\frac{\pi}{n} \csc \left(\frac{\pi}{n}\right)\left[... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{{\displaystyle #1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4577083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Find All Complex solutions for $z^3+3i\bar{z}=0$ Find All Complex solutions for $z^3+3i \bar z =0$.
I tried substituting $z=a+bi$ and simplifying as much as I can
and this is what I ended up with: $a^3-2b^2a+3b+i(3ba^2-b^3+3a)=0$.
I just did not understand how do I get the values of $z$ from this equation
| You let $z=a+bi$ and the system of equations to solve is: $a^3-3ab^2+3b=0$ and $3ba^2-b^3+3a=0$. If you multiply the first equation by $-a$ and the second by $b$ and then add the resulting equations you get $a^4+b^4=6a^2b^2$. Let $m=\frac{b}{a}$. Then, $$m^4-6m^2+1=0.$$ The solution of this equation is: $m=\tan(\frac{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4578068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How far can we go with the integral $I_n=\int_0^1 \frac{\ln \left(1-x^n\right)}{1+x^2} d x$ Inspired by my post, I decided to investigate the integral in general
$$
I_n=\int_0^1 \frac{\ln \left(1-x^n\right)}{1+x^2} d x$$
by the powerful substitution $x=\frac{1-t}{1+t} .$
where $n$ is a natural number greater $1$.
Let’s... | As @FShrike’s solution, I want to express the integral in terms of an infinite series of Diagamma functions.
Using $\ln \left(1-x^n\right)=-\sum_{k=1}^{\infty} \frac{x^{n k}}{k}$ for $|x|<1$, we have
$$
\int_0^1 \frac{\ln \left(1-x^n\right)}{1+x^2} d x=-\sum_{k=1}^{\infty} \frac{1}{k} \underbrace{\int_0^1 \frac{x^{n k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4579985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
What is the solution for $x+22=-6\sqrt{2x+9}$ So, I want to solve for x in the radical equation:
$x + 22 = -6\sqrt{2x+9}$
By Squaring each expression we get:
$(x + 22)^2 = (-6\sqrt{2x+9})^2$
$ x^2 + 44x + 484 = 36\cdot(2x+ 9) $
$ x^2 + 44x + 484 = 72x + 324 $
Now by solving the quadratic equation:
$ x^2 - 28x + 160 = ... | Sometimes it can be useful to find the domain of the equation before solving equations involving radicals.
By definition of the principal square root, if $\sqrt {f(x)}=g(x)$, then $f(x)\geqslant0\wedge g(x)\geqslant 0$ holds, where $f(x)$ and $g(x)$ are some algebraic expressions.
Thus, you have:
$$
\begin{align}&\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4580801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Literature containing the process of solving $ f''(x)+ae^{bx}f(x)=0 $ I tried to solve the following ODE using infinite summs, but failed:
$$ f''(x)+ae^{bx}f(x)=0 \tag1$$
From a comment in this post, I found out that the solution of the above equation contains a linear combination of the Bessel functions. That means th... | Consider the differential equation:
$$ \frac{d^2 y}{d x^2} + a^2 \, e^{b x} \, y = 0. $$
First note that if $a = 0$ then $y(x) = c_{0} + c_{1} \, x$. Second note that if $b = 0$ then $f(x) = c_{0} \, \cos(a x) + c_{1} \, \sin(a x)$. Now consider the case when $a \neq 0$ and $b \ne 0$, which is as follows.
Let $t = e^{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4582183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to solve $\cos^{40}x-\sin^{40}x=1$ for real and imaginary solutions Solve : $$\cos^{40}x-\sin^{40}x=1$$
I found by induction that $\sin(x)=0$ satisfy the equation so $$x=n\pi$$ must be the solution but there should be more solutions real or imaginary , how to find them ?
| Notice that, with $t = \cos x$ and $y = \sin x$, the equation factors into the form
\begin{align}
1 &= (t - y) (t + y) (t^2 + y^2) (t^4 + y^4) (t^4 - t^3 y + t^2 y^2 - t y^3 + y^4) \\
& \hspace{5mm} \cdot (t^4 + t^3 y + t^2 y^2 + t y^3 + y^4) (t^8 - t^6 y^2 + t^4 y^4 - t^2 y^6 + y^8) \\
& \hspace{5mm} \cdot (t^{16} - t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4583530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Prove that $(x^2−1) \bmod 8$ $\in \{ 0 , 3 , 7 \}, \forall x \in \mathbb{Z}$. It must be verified that for all $x \in \mathbb{Z}$ it holds that $x^2 - 1 \bmod{8} \in \{0, 3, 7\}$. First some definitions. Using the following theorem a definition for $\bmod$ is provided:
Theorem. For all $a \in \mathbb{Z}$ and $d \in \m... | $$x^2-1=(x-1)(x+1)$$
If $x$ is even, $x-1$ and $x+1$ are both odd. The possibilities are then $7\times1$, $1\times3$, $3\times5$ and $5\times7$, which gives you $3$ or $7$.
If $x$ is odd, $x-1$ and $x+1$ are both even, and one of them is divisible by $4$, so the remainder is always $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4584496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Prove that $\operatorname{lcm}$ of [$\binom{n}{1}$, $\binom{n}{2}$, ... ,$\binom{n}{n}$] = $\operatorname{lcm}(1, 2, ...,n+1)/(n+1)$ How to prove that:
$$\operatorname{lcm}\left(\binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n}\right) = \frac{\operatorname{lcm}(1, 2, \ldots, n+1)}{n+1}$$
There is a hint:
$p$ is a prim... | Lemma: $[\frac{n}{p}]-[\frac{n-k}{p}]-[\frac{k}{p}]\le 1$
Note that $$v_p(\binom{n}{k})=\sum_{i=1}^{\infty} [\frac{n}{p^i}]-[\frac{n-k}{p^i}]-[\frac{k}{p^i}]$$
Suppose $p^e\le n+1 <p^{e+1}$ for some $p<n$, then $v_p(\operatorname{lcm}(1, ..., n+1))=e$.
We need to prove that
$$(n+1)v_p(\operatorname{lcm}(\binom{n}{1}, .... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4586202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
If $2^\frac{2x-1}{x-1}+2^\frac{3x-2}{x-1}=24$, find all values of $x$ that satisfy this As title suggests, the problem is as follows:
Given that $$2^\frac{2x-1}{x-1}+2^\frac{3x-2}{x-1}=24$$ find all values of $x$ that satisfy this.
This question was shared in an Instagram post a few months ago that I came across toda... | First of all, your solution looks great.
Here's a slightly different way to discover that $x=2$. Noticing the $x-1$ denominators in the exponents, I would start by making the substitution $u = x - 1$, so $x = u + 1$. Thus,
$$
\frac{2x-1}{x-1} = \frac{2u+1}{u} = 2 + \frac{1}{u}
$$
and
$$
\frac{3x-2}{x-1} = \frac{3u+1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4587013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Why i calculate this limit like this? Calculate $ \lim_{x \to ∞} f(x) = ($$\frac{x^2-2x+7}{x^2+3x+1})^\frac{1}{\sin(1/x)}$
The solution involves using $ \lim_{t \to 0} (1+t)^\frac{1}{t}=e $
But can i say that if $ \lim_{x \to ∞} ($$\frac{x^2-2x+7}{x^2+3x+1})$ = 1
and $ \lim_{x \to ∞}\frac{1}{\sin(1/x)} = 1 $
then $ \... | Assuming the limit exists, let
$\displaystyle L:= \lim_{x \to +\infty} \left(\frac{x^2-2x+7}{x^2+3x+1}\right)^\frac{1}{\sin(1/x)}$
Then we have:
$$\log L=\lim_{x \to +\infty} \frac{1}{\sin(1/x)} \log \left(\frac{x^2-2x+7}{x^2+3x+1}\right)$$
Since
$$\sin \left(\frac{1}{x}\right) \sim \frac{1}{x} \quad \text{as}\quad x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4594328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
I got a different value with wolfram of evaluating $~\int_{0}^{2\pi}\left|\sin\theta+\cos\theta\right|\mathrm d \theta$ $$\begin{align}
A&:=\int_{0}^{2\pi}\left|\sin\theta+\cos\theta\right|\mathrm{d}\theta\\
\end{align}$$
$$\begin{align}
\sin\theta+\cos\theta&<0\leftrightsquigarrow\theta\neq{\pi\over2},{3\pi\over 2}\\
... | I would like to evaluate the integral from definition of absolute value by splitting the integration interval into 3 intervals.
$$
\begin{aligned}
\int_0^{2 \pi}|\sin \theta+\cos \theta| d \theta
\stackrel{\theta\mapsto\theta-\frac{\pi}{4}}{=}& \sqrt{2} \int_0^{2 \pi}\left|\cos \left(\theta-\frac{\pi}{4}\right)\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4601049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Find all three complex solutions of the equation $z^3=-10+5i$ Let $z\in \mathbb{C}$. I want to calculate the three solutions of the equation $z^3=-10+5i$. Give the result in cartesian and in exponential representation.
Let $z=x+yi $.
Then we have $$z^2=(x+yi)^2 =x^2+2xyi-y^2=(x^2-y^2)+2xyi$$
And then $$z^3=z^2\cdot z=[... | Use a primitive third root of unity, say $\zeta_3=e^{\frac {2\pi i}3}.$
Take $\omega $ with $w^3=-10+5i$, say $\omega =-\sqrt {125}^{\frac 13}e^{\frac{i\arctan -\frac 12}3}=-\sqrt 5e^{\frac{i\arctan -\frac 12}3}.$
Then the roots are $\{\omega, \zeta_3 \omega, \zeta_3 ^2\omega \}.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4601201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Draw an area on the complex plane On the complex plane draw the area:
$$
\begin{equation}
\begin{cases}
|z+4i| < 3 \\
|\arg(z-5-5i)|<\frac{\pi}{3}
\end{cases}
\end{equation}
$$
Where $ \arg(z) \in (-\pi, \pi ]$
I can draw $|z+4i| < 3$:
$|x + iy + 4i|<3 \Rightarrow \sqrt{x^2 + (y + 4)^2}<3 \Righta... | You must intersect the circle with:
$$|arg(z-5-5i)|<\frac{\pi}{3}$$
which means:
$$\left| \arctan \left(\frac{\Im(z - 5 - 5i)}{\Re(z - 5 - 5i)} \right) \right | < \frac{\pi}{3}$$
$$\Longleftrightarrow \Re \left\{\arctan \left(\frac{\Im(z) - 5}{\Re(z) - 5} \right)^2 \right\} + \Im \left\{\arctan \left(\frac{\Im(z) - 5}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4602374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Does $\sum\limits_{n = 1}^{\infty} \frac{3^n + 4^n}{2^n + 5^n}$ converge? My Attempt
First, check the limit
$$\lim_{n \to \infty} \frac{3^n + 4^n}{2^n + 5^n} =
\lim_{n \to \infty} \frac{\left(\frac{3}{5}\right)^n + \left(\frac{4}{5}\right)^n}{\left(\frac{2}{5}\right)^n + 1} = 0.$$
So, we cannot conclude anything.
I us... | $$\left(\frac{2}{5}\right)^n + 1 > 1 $$
Hence
$$\frac{\left(\frac{3}{5}\right)^n + \left(\frac{4}{5}\right)^n}{\left(\frac{2}{5}\right)^n + 1} < \left(\frac{3}{5}\right)^n + \left(\frac{4}{5}\right)^n $$
And both $\left(\frac{3}{5}\right)^n$ and $\left(\frac{4}{5}\right)^n$ are general terms of a convergent geometric s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4603874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Sum of two subspaces: representing it with equations I found the following excercise:
Let $W_1 = \{(x_1, ..., x_6) : x_1 + x_2 + x_3 = 0, x_4 + x_5 + x_6 = 0 \}$. Let $W_2$ be the span of $S := \{(1, -1, 1, -1, 1, -1), (1, 0, 2, 1, 0, 0), (1, 0, -1, -1, 0, 1), (2, 1, 0, 0, 0, 0)\}$.
Give a base, a dimension and an eq... | You already found that a vector $(a,b,c,d,e,f)$ belongs to $W_1+W_2$ if and only if there exist real numbers $x_2,x_3,x_5,x_6,x,y,z,w$ such that
$$\begin{cases}a&=x+y+z+2w-x_2-x_3\\b&=-x+w-x_2\\c&=x + 2y - z + x_3\\
d&=-x + y - z-x_5 - x_6\\
e&=x + x_5\\
f&=-x + z + x_6.
\end{cases}$$
Using for instance the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4605954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$3\times3$ determinant using standard basis I am trying to get from a $2\times2$ determinant to a $3\times3$ determinant.
$$\left|\begin{array}{c1 c2 c3}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
$$
How does one get to
$$ \det(A)=\sum_{j=1}^3 a_{j1} \; \de... | Consider the given determinant as $\begin{vmatrix}\mathbf{a}_1 & \mathbf{a}_2 &\mathbf{a}_3 \end{vmatrix}$, where $\mathbf{a}_j$ is the $j-$th column.
By multi-linearity of the determinants (in particular keeping the second and third columns the same), we get
$$\begin{vmatrix}\color{red}{\mathbf{u} +\lambda\mathbf{b}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4607157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Recurrence Relation in Reduction formula for Integral $\sin^n(x)$ For the integral $\displaystyle \int\sin^n(x) dx$ there exists the following reduction formula, that is a recurrence relation:
$\displaystyle I_n = \frac{n-1}{n} \cdot I_{n-2}-\frac{\sin^{n-1}(x) \cdot\cos(x)}{n}$
I have now been trying to solve this re... | After separating even and odd terms of the sequence, i.e.
$$
\begin{cases}
\displaystyle
a_n := I_{2n} = \frac{2n-1}{2n}I_{2n-2} - \frac{\sin^{2n-1}(x)\cos(x)}{2n} = \frac{2n-1}{2n}a_{n-1} - \frac{\sin^{2n-1}(x)\cos(x)}{2n} \\
\displaystyle
b_n := I_{2n+1} = \frac{2n}{2n+1}I_{2n-1} - \frac{\sin^{2n}(x)\cos(x)}{2n+1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4610074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How do we prove $x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0$? Question
How do we prove the following for all $x \in \mathbb{R}$ :
$$x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0 $$
My Progress
We can factorise the left hand side of the desired inequality as follows:
$$x^6+x^5+4x^4-12x^3+4x^2+x+1=(x-1)^2(x^4+3x^3+9x^2+3x+1)$$
However, after... | $$
x^4+3x^3+9x^2+3x+1 = x^2\left(x+\frac 32\right)^2 + \frac{27}{4}\left(x+\frac 2 9\right)^2 + \frac 23
$$
is strictly positive for all real $x$.
How did I come up with that? I started by completing the square in
$$
x^4+3x^3+9x^2 = x^2(x^2+3x+9) = x^2\left( (x+\frac 32)^2 + \frac{27}{4}\right)
$$
so that
$$
x^4+3x^3+9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4611737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 0
} |
If $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$, show that $a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab} \leq abc$. If $a$, $b$, $c$ are nonzero natural numbers and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$, show that $a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab} \leq abc$.
I basically have no idea how I can start this problem.
I thought... | Multiply $abc$ in $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ to get:$$bc+ac+ab=abc$$Now, by AM-GM: $$a\sqrt{bc}+b\sqrt{ac}+c\sqrt{ab}\le\frac{1}{2}(a(b+c)+b(a+c)+c(a+b)=\frac{1}{2}(ab+ac+ab+bc+ac+bc)=ab+bc+ac=abc$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4612335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
determine the product of all possible values of $|a+b+c|$
Let $a,b,c$ be complex numbers such that $|a|=|b|=|c|=1$. If $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}=1$ as well, then find the product of all possible values of $|a+b+c|$.
Suppose we know that $s^3 = abc(3|s|^2-2),$ where $s = a+b+c.$ Then $|s|$ is a... | As a different way which shows why the values are obtainable:
Let's assume $a=e^{i \theta_1}, b=e^{i \theta_2}$, and $c=e^{i \theta_3}$. Then, because of $\frac{a^2}{bc}+ \frac{b^2}{ac}+\frac{c^2}{ab}=1$, we must have:
$$\cos (2\theta_1-\theta_2-\theta_3)+\cos (2\theta_2-\theta_1-\theta_3)+\cos (2\theta_3-\theta_2-\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4613576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve the equation $\log_{1-2x}(6x^2-5x+1)-\log_{1-3x}(4x^2-4x+1)=2$ Solve the equation $$\log_{1-2x}(6x^2-5x+1)-\log_{1-3x}(4x^2-4x+1)=2$$
We have $$D_x:\begin{cases}1-2x>0\\6x^2-5x+1>0\\1-3x>0\\1-3x\ne1\\4x^2-4x+1>0\iff(2x-1)^2>0\iff x\ne\dfrac12\end{cases}\iff x\in(-\infty;0)\cup(0;\dfrac{1}{3})$$
Also the quadratic... | The key here is to use the change of base formula for logs: $\log_c(x)=\frac{\log_b(x)}{\log_b(c)}$. Using this allows us to express everything in terms of a common base, which then lets us use other log rules.
$$\frac{\log (1-2x)(1-3x)}{\log (1-2x)}-\frac{\log[(1-2x)^2]}{\log(1-3x)}=2.$$
Using log properties, this si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4615148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Simplifying $\frac{2x^3-9x^2+27}{3x^3-81x+162} $ Simplify $$\frac{2x^3-9x^2+27}{3x^3-81x+162} $$
All I can see is thus far is the factor 3 in the denominator can be taken out. Then I am stuck because I don't recognize any of the usual patterns in simplification.
The answer is $\;\dfrac{2x+3}{3(x+6)}\;,\;$ so clearly I ... | By the rational roots theorem, the rational roots of the numerator are $-\frac32$ and $3$ and, if you divide the numerator by $\left(x+\frac32\right)(x-3)$, you will get $2x-6$. Therefore, the numerator is equal to $(2x+3)(x-3)^2$. On the other hand, the rational roots of the denominator are $-6$ and $3$; in fact, the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4617155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Showing $x_n-x_nx_1+\sum_{k=1}^{n-1} (x_k-x_kx_{k+1})\leq\left\lfloor\frac{n}{2}\right\rfloor$, with $x_i\in[0,1]$
Let $x_1, x_2,\ldots, x_n$ be arbitrary numbers from the interval $[0,1]$ with $n>1$.
Show that $$x_n-x_nx_1+\sum_{k=1}^{n-1} (x_k-x_kx_{k+1})\leq\left\lfloor\frac{n}{2}\right\rfloor$$
I tried to factor ... | When $n$ is even, we have
\begin{align*}
x_1(1-x_2) + x_2(1-x_3)
= 1 - (1 - x_1)(1 - x_2) - x_2 x_3 &\le 1,\\
x_3(1-x_4) + x_4(1 - x_5) = 1 - (1-x_3)(1-x_4) - x_4x_5 &\le 1,\\
\cdots \cdots \cdots \cdots &\qquad\\
x_{n-1}(1 - x_n) + x_n(1 - x_1)
= 1 - (1 - x_{n-1})(1 - x_n) - x_nx_1 &\le 1.
\end{align*}
Adding t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
How to derive a closed form of a recursion (maybe using generating functions) Let $a_0=9$ and consider the following recurrence relation: $$a_n=36(n+1)2^{n-2}+2a_{n-1},$$ I'm looking for the closed form of $\{a_n\}.$
I have tried using generator functions:
\begin{align*}
f(x)&=\sum_{n=0}^\infty a_nx^n\\
&=9 +\sum_{n=1}... | An easy way to solve this recurrence.
As the recurrence is linear, we have that
$$
a_n = a_n^h + a_n^p
$$
such that
$$
\cases{a_n^h= 2 a_{n-1}^h\\ a_n^p = 2 a_{n-1}^p+36(n+1)2^{n-2}}
$$
so $a_n^h = c_0 2^n$. Now assuming $a_n^p = c_0(n)2^n$ after substitution into the complete equation, we have
$$
c_0(n)2^n = 2c_0(n-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solve the equation $8^x+3\cdot2^{2-x}=1+2^{3-3x}+3\cdot2^{x+1}$ Solve the equation $$8^x+3\cdot2^{2-x}=1+2^{3-3x}+3\cdot2^{x+1}$$ The given equation is equivalent to $$2^{3x}+\dfrac{12}{2^x}=1+\dfrac{8}{2^{3x}}+6\cdot2^x$$ If we put $a:=2^x>0$, the equation becomes $$a^3+\dfrac{12}{a}=1+\dfrac{8}{a^3}+6a$$ which is $$a... | Letting $y=2^{x-\frac12}>0$
$$\begin{align}8^x+3\cdot2^{2-x}=1+2^{3-3x}+3\cdot2^{x+1}&\iff y^3+\frac3y=\frac1{2\sqrt2}+\frac1{y^3}+3y
\\&\iff\left(y-\frac1y\right)^3=\left(\frac1{\sqrt2}\right)^3\\&\iff\dots\\&\iff y=\sqrt2\\&\iff x=1.\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4619652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Compute $\int_a^b e^x dx$ as a Riemann Sum I tried computing the integral $$\int_a^b e^x dx$$ as a Riemann sum. Therefore split the interval in to $n$ parts of the length $$\frac{b-a}{n}$$
and then took the limit of the Riemann sum.
$$\lim _{n \rightarrow \infty} \frac{b-a}{n} \sum_{k=0}^n e^{\frac{k(b-a)}{n}}$$.
When ... | $$
\begin{aligned}
\int_a^b e^x d x & =\lim _{n \rightarrow \infty} \frac{b-a}{n} \sum_{k=0}^n e^{a+\frac{k(b-a)}{n}} \\
& =e^a \lim _{n \rightarrow \infty} \frac{b-a}{n} \sum_{k=1}^n\left(e^{\frac{b-a}{n}}\right)^k \\
& =e^a \lim _{n \rightarrow \infty} \frac{b-a}{n} \cdot \frac{\left(e^{\frac{b-a}{n}}\right)^n-1}{e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proving $\left\lfloor(\frac{1+\sqrt{5}}{2})^{4n+2}\right\rfloor-1$ is a perfect square for $n=0,1,2,\ldots$ Let $$S_n = \left \lfloor\left(\frac{1+\sqrt{5}}{2}\right)^{4n+2}\right\rfloor-1$$ ($n=0, 1, 2, \ldots$).
Prove that $S_n$ is a perfect square.
In Art of Problem Solving website, there is a hint
$$
\begin{align}
... | Let $\varphi:=\frac{1+\sqrt5}2$ and $\bar\varphi:=\frac{1-\sqrt5}2.$ The hint you wonder about,
$$\left\lfloor\varphi^{2m}\right\rfloor=\varphi^{2m}+\bar\varphi^{2m}-1,$$
is due to the fact that $\varphi^k+\bar\varphi^k$ is an integer (the $k$-th Lucas number) and
$$\varphi^{2m}+\bar\varphi^{2m}-1\le\varphi^{2m}<\varph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Inverse of difference of two digamma functions I recently encountered the expression below for which I was interested in solving for $x$:
\begin{equation}
\psi(x+n+1) - \psi(x+1) =y
\end{equation}
$\psi$ is the digamma function, $n$ is a positive integer and $x,y>0$. Initially, I solved this numerically but I was inter... | $$\psi(x+n+1) - \psi(x+1) =y$$
For large values of $x$, the lhs write
$$\frac{n}{x}-\frac{n (n+1)}{2 x^2}+\frac{n (n+1) (2 n+1)}{6
x^3}-\frac{n^2 (n+1)^2}{4
x^4}+O\left(\frac{1}{x^5}\right)$$ Using series reversion
$$x_{(1)}=\frac{n}{y}-\frac{n+1}{2}+\frac{\left(n^2-1\right)}{12
n}y+O\left(y^3\right)$$
Compari... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $7 \mid 11^n - 4^n$ with mathematical induction I want to prove that $7 \mid 11^n - 4^n$ with mathematical induction. This is what I wrote:
*
*For $n = 1$, we have $7 \mid 11^1 - 4^1 \Rightarrow 7 \mid 7$ which is obviously true. $\checkmark$
*Assume that the statement is true for $n = k$. So: $7 \mid 11... | Let's modify your induction proof so that we directly use the induction hypothesis that $7 \mid 11^k - 4^k$ for some positive integer $k$.
Let $P(n)$ be the the statement that $7 \mid 11^n - 4^n$.
You handled the $n = 1$ case correctly.
Since the statement $7 \mid 11^n - 4^n$ holds for $n = 1$, we may assume $P(k)$ hol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Nth Derivative of $\frac{x}{x-1}$ So starting off, rewrite $x*\dfrac1{x-1}$. $d/dx=\dfrac{d}{dx}(x)*\dfrac{1}{x-1}+\dfrac{d}{dx}\dfrac{1}{x-1}*x$.
$\dfrac{d}{dx}(x)=1$. $x^{n}=nx^{n-1}$.
$\dfrac{1}{x-1}=(x-1)^{-1}$.
$\dfrac{d}{dx}(\dfrac{1}{x-1})=(-1)(x-1)^{-2}=\dfrac{-1}{(x-1)^{2}}$.
Combining them both we get $\dfrac... | How about
$$f(x)=\frac x{x-1}=\frac1{x-1}+1$$
Therefore:
$$\frac d{dx}f(x)=-1\frac1{(x-1)^2}\\\frac{d^2}{dx^2}f(x)=-1\cdot-2\frac1{(x-1)^3}\\\frac {d^3}{dx^3}f(x)=-1\cdot -2\cdot -3\frac1{(x-1)^4}\\\vdots\\\frac{d^n}{dx^n}f(x)=\frac{(-1)^n n!}{(x-1)^{n+1}}$$
One may use factorial power $u^{(v)}$ to get:
$$\boxed{\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Integration $\int_0^{\pi/2} \frac{dx}{(3 + 5 \cos x)^2}$ I had tried to solve this integral; using the substitution $\tan(x/2) =t$, and $\cos x= \frac{1-t^2}{1+t^2}$. But after making terms in $t$, I am not able to integrate further as numerator contains quadratic and denominator contains biquadratic.
$\int\limits_0^{\... | Well let's use Weierstrass and partial fractions ig and evaluate the definite with FTC II.
Using Weierstrass with the tangent half angle substitution, our integral becomes
$$\int \frac{1}{(3 + 5 \cos x)^2}\ dx = \int \frac{1}{\left(3+5\left(\frac{1-t^2}{1+t^2}\right)\right)^2}\cdot \frac{2dt}{1+t^2} = \frac12\int {t^2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4622458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Calculate the limit of a function involving greatest integer function
If $f:(0,\infty)\rightarrow\mathbb{N}$ and
$$f(x)=\left[\frac{x^2+x+1}{x^2+1}\right]+\left[\frac{4x^2+x+2}{2x^2+1}\right]+\left[\frac{9x^2+x+3}{3x^2+1}\right]+\cdots+\left[\frac{(nx)^2+x+n}{nx^2+1}\right]$$
Find the value of $$\lim_{n\rightarrow\inf... | $f(x)$ is easy to calculate once you simplify the algebraic expressions under it.
$f(x)=\begin{equation}
\sum_{k=1}^{n}\left[\frac{(nx)^2+x+n}{nx^2+1}\right]
\end{equation}$
$\left[\frac{(nx)^2+x+n}{nx^2+1}\right] = \left[\frac{(nx)^2+n}{nx^2+1}+\frac{x}{nx^2+1}\right]$
$=\left[\frac{n(nx^2+1)}{nx^2+1}+\frac{x}{nx^2+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4624216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
All natural number solutions for the equation $a^2+b^2=2c^2$ $a$, $b$ and $c$ of all Pythagorean triplets can be written in the form
$$
\begin{split}
a &= 2mn\\
b &= m^2-n^2 \\
c &= m^2+n^2
\end{split}
$$
where $m$ and $n$ are natural numbers. For any natural number $m$ and $n$, this set of equations will give a Pythag... | Noting that
$a^2+b^2=2c^2 \Rightarrow a $ and $b$ are of same parity. Hence there exists natural numbers $u$ and $v$ such that
$$a+b=2u \textrm{ and } a-b=2v$$
Then $$a=u+v \textrm{ and }b=u-v$$
$$
\begin{gathered}
(u+v)^2+(u-v)^2=2 c^2 \Rightarrow
u^2+v^2=c^2
\end{gathered}
$$
There exists Pythagorean triple such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4624896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find the radical centre of the four spheres :$x^2+y^2+z^2+2x+2y+2z+2=0,x^2+y^2+z^2+4y=0,x^2+y^2+z^2+3x-2y+8z+6=0,x^2+y^2+z^2-x+4y-6z-2=0$ Determine the radical centre of the spheres $x^2+y^2+z^2+2x+2y+2z+2=0,x^2+y^2+z^2+4y=0,x^2+y^2+z^2+3x-2y+8z+6=0,x^2+y^2+z^2-x+4y-6z-2=0.$
I tried solving the problem by assuming $S_1... | Here is the computation.
$\quad\quad$ $(x,y,z)$ is the radical centre
$\iff S_1(x,y,z)=S_2(x,y,z)=S_3(x,y,z)=S_4(x,y,z)$
$\iff 2x+2y+2z+2=4y=3x-2y+8z+6=-x+4y-6z-2$
$\iff x=y-z-1\text{ and }x=2y-\frac83z-2=0\text{ and } x=-6z-2$
$\iff x=y-z-1\text{ and }y-\frac53z-1=0\text{ and } y+5z+1=0$
$\iff x=-\frac15, y=\frac12, z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4628883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the limit $\lim_{x\to2}(x^3-2x-4)\tan\frac{\pi x}{4}$ Find the limit $$\lim_{x\to2}(x^3-2x-4)\tan\dfrac{\pi x}{4}$$
I can't even determine what type of indeterminate form we have. It's $0\times\text{undefined}$. We can see that $2$ is a root of the polynomial $x^3-2x-4$ and it factors as $(x-2)(x^2+2x+2)$. Additio... | We have:
$$ (x^3-2x-4) \tan(x\pi/4) = \frac{\sin(x\pi/4)(x-2)(x^2+2x+2)}{\cos(x\pi/4)}$$
Let $u = x-2$, then:
$$ = \frac{u\sin((u+2)\pi/4)((u+2)^2+2(u+2)+2)}{\cos((u+2)\pi/4)} $$
Where $\sin((u+2)\pi/4) = \cos(u\pi/4)$ and $\cos((u+2)\pi/4) = -\sin(u\pi/4)$ using the addition formulas for sine and cosine. So:
$$ \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I find individual values of $\sin(\varphi)$ and $\cos(\varphi)$ from $x = a\sin^{2}(\varphi) + b\cos^{2}(\varphi)$? If $x = a\sin^2\phi+ b\cos^2\phi$, express $\sin$ and $\cos$ in terms of $x$ ( $a$ and $b$ are real constants)
I know how to find values of $T$ ratios in equations like $x = a\sin^2t$ or $x = b\cos... | 1. $$x=a\sin^2\phi + b\cos^2\phi$$
2. $$x=a\sin^2\phi+b-b\sin^2\phi$$
3. $$x=b+(a-b)\sin^2\phi$$
4. $$\sin\phi=\pm\sqrt\frac{x-b}{a-b}$$
$$\cos\phi=\pm\sqrt\frac{a-x}{a-b}$$
Hope this helps!!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solve $(x^2+1)(x-1)^2=ax^2, \ a\in\mathbb{R}$
Solve $(x^2+1)(x-1)^2=ax^2, \ a\in\mathbb{R}$. Discuss.
This is the entire text of the problem. I am really unsure what the authors meant by discuss here? Is it to find $x$ based on $a$ and then separate by cases? What I tried doing is to raise to power and multiply the p... | $(x^2+1)(x-1)^2=ax^2\quad\color{blue}{(*)}$
Since $\,x=0\,$ is not a solution of the equation $(*)\,,\,$ we can divide by $\,x^2$ both sides of $(*)$ and get the following equivalent equation:
$\left(x+\dfrac1x\right)\left(x+\dfrac1x-2\right)=a\;.$
By letting $\;t=x+\dfrac1x\,,\,$ we get the following equation:
$t\,(t-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4632973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Without Calculator find $\left\lfloor 2 \cos \left(50^{\circ}\right)+\sqrt{3}\right\rfloor$
Without Calculator find $$\left\lfloor 2 \cos \left(50^{\circ}\right)+\sqrt{3}\right\rfloor$$
Where $\left \lfloor x \right \rfloor $ represents floor function.
My Try:
Let $x=2\cos(50^{\circ})+\sqrt{3}$. We have
$$\begin{al... | 50° is between 45° and 60° ( $50°=45°+(60°-45°)/3$ ), and in this range, function $cos$ is concave. So $\cos(50°) > \cos 45°+(\cos60°-\cos45°)/3$
$\cos 50° > \sqrt{2}/2 + (1/2- \sqrt{2}/2)/3$.
Approximate $\sqrt{2}$ with $1.414$ and $\sqrt{3}$ with $1.732$, and you are in.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4633391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
Find the roots of the equation $(1+\tan^2x)\sin x-\tan^2x+1=0$ which satisfy the inequality $\tan x<0$ Find the roots of the equation $$(1+\tan^2x)\sin x-\tan^2x+1=0$$ which satisfy the inequality $$\tan x<0$$
Shold I solve the equation first and then try to find which of the roots satisfy the inequality? Should I use ... | $\frac\pi2+2k\pi$ should never appear because the initial equation is not defined when $\cos x=0.$
You forgot a $-$ sign in front of $\frac{5\pi}6+2k\pi.$
As for your final question: simply discard $-\frac{5\pi}6+2k\pi$ (whose $\tan$ is $>0$) and retain $-\frac\pi6+2k\pi$ (whose $\tan$ is $<0$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4639727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Keeping track of basis changes when computing the smith normal form I don't understand how to keep track of the basis change as you compute smith normal form. We did an example in lecture where:
$A = \begin{pmatrix}
[1] & 2 & 3 & 4 \\
5 & 6 & 7 & 8 \\
9 & 10 & 11 & 12 \\
\end{pmatrix}$
Where [1] is the pivot position. ... | You can think of the decomposition process as follows: starting with $A_0 = A$, $P_0 = I_3$, and $Q_0 = I_4$, we go from $(A_k,P_k,Q_k)$ to $(A_{k+1},P_{k+1},Q_{k+1})$ in such a way that at all points in the process, we have $A = P_k A_k Q_k$. Note that $P_k$ is a change of basis matrix over the codomain and $Q_k$ is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4640462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int \sqrt{\frac{1+x^2}{x^2-x^4}}dx$
Evaluate $$\large{\int} \small{\sqrt{\frac{1+x^2}{x^2-x^4}} \space {\large{dx}}}$$
Note that this is a Q&A post and if you have another way of solving this problem, please do present your solution.
| In order to evaluate the integral over all domain $ x\in (-1,0)\cup(0,1)$, substitute $t=x\sqrt{x^2}$. Then, $dt =2\sqrt{x^2}\ dx$ and
\begin{align}
\int \sqrt{\frac{1+x^2}{x^2-x^4}}\ dx
=& \int \frac{1+x^2}{\sqrt{x^2(1-x^4)}}\ dx
=\frac12\int \frac{1+\sqrt{t^2}}{\sqrt{t^2(1-t^2)}}\ dt \\
=& \ \frac12\int \frac{1}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4641473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Given such $n$th degree polynomial $P(x)$ and positive numbers $a, b\in\mathbb{R}$, does $\sqrt[n]{P(a+b)} \leq \sqrt[n]{P(a)} + \sqrt[n]{P(b)}$ hold? Given any $n$th degree polynomial $P(x)$ with positive coefficients and positive numbers $a, b\in\mathbb{R}$, does $\sqrt[n]{P(a+b)} \leq \sqrt[n]{P(a)} + \sqrt[n]{P(b)}... | From https://artofproblemsolving.com/community/c6h1440215p8188094 on AoPS:
If $P(x) = \sum_{k=0}^n c_k x^k$ is a polynomial with nonnegative coefficients and $a, b \ge 0$ then
$$
\begin{align}
\sqrt[n]{P(a+b)} &= \left( \sum_{k=0}^n c_k (a+b)^k\right)^{1/n}\\
&\overset{(1)}{\le} \left(\sum_{k=0}^n c_k( a^{k/n} + b^{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4645516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Convergence of infinite product and its limit I wanted to find $\prod_{n=2}^{\infty}(1+\frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+...)$ and ended up simplifying it as
$\prod_{n=2}^{\infty} \frac{n^2}{n^2-1}$. Now the partial product is $\frac{2n}{n+1}$ it converges and it's limit is 2.. am I correct? Kindly share your v... | We can check your results, in order to verify your answer.
*
*First, we can show that
$$1+\frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots =\sum_{k=0}^{+\infty}\frac{1}{n^{2k}}.$$
*If $n\geqslant 1$, we can show that
$$\sum_{k=0}^{+\infty}\frac{1}{n^{2k}}=\sum_{k=0}^{+\infty}\left(\frac{1}{n^2}\right)^k=\frac{n^2}{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4648058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the number of ways to arrange so that at least two are followed
Suppose there is a sentence containing only sequences of three characters and nothing more. The three characters are $X,Y,Z$ and it is given that $X$ has occurred $a$ times, $Y$ has occurred $b$ times and $Z$ has occurred $c$ times in the sentence. W... | As this answer has now been accepted, I should mention that it’s unnecessarily complicated and the stars-and-bars argument provided by Daniel Mathias in a comment below is much more elegant.
You can proceed as in this nice answer by A.J. to What is the probability of 2 named cards appearing sequentially in a randomly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4648839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Valid proof for integral of $1/(x^2+a^2)$ I'm trying to prove some integral table formulae and had a concern over my proof of the following formula:
$$\int\frac{1}{x^2+a^2}\;dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$$
Claim:
$$\frac{1}{x^2+a^2}=\frac{1}{a^2}\sum_{k=1}^\infty(-1)^{k-1}\left(\frac{x}{a}\right)^{2k}... | It’s valid for $|x|\leq 1$ due to the uniform convergence of the Taylor series you obtain after integrating to arctan, but not outside this interval and has to do with not being able to pull that sum outside the integral on this region. I can give you a simpler way to integrate this if you’d like?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4651435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Difficulties with estimation of epsilon-delta limit proof I have to proof $\lim_{x\to 5}\frac{4x-9}{3x-16}=-11$. I have hard time to evaluate $\frac{1}{|3x-16|}$.
So I start that let $|x-5|<$. I need to show that $|\frac{4x-9}{3x-16}-(-11)|<ε$.
$|\frac{4x-9}{3x-16}-(-11)|$ = $|\frac{37x-185}{3x-16}|$ = $\frac{5|x-5|}{|... | A systematic advice in such a situation is to "translate the variable to $0$", i.e. here: let
$$x=5+h,$$
so that $x\to5\iff h\to0.$ If $x\ne\frac{16}3,$
$$\frac{4x-9}{3x-16}+11=\frac{37h}{-1+3h}.$$
If $h<\frac13$ then
$$\begin{align}\left|\frac{37h}{-1+3h}\right|<\varepsilon&\iff37|h|<\varepsilon(1-3h)\\&\Longleftarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4651858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How can you prove that the square root of two is irrational? I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational?
| Here's a short algebraic proof. It nowhere uses rules about primes or even numbers.
You need to first show that $1<\sqrt{2}<2,$ but that is obvious.
We first assume that $\sqrt{2}$ is rational. Then pick the smallest positive $q$ so that $p=q\sqrt{2}$ is an integer. Then $q<p<2q.$
Now compute:
$$\left(\frac{2q-p}{p-q}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "64",
"answer_count": 17,
"answer_id": 3
} |
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,... | Let's say you have a sequence of nonnegative numbers $a_1 \geq a_2 \geq \dots$ tending to zero. Then it is a theorem that the alternating sum $\sum (-1)^i a_i$ converges (not necessarily absolutely, of course).
This in particular applies to your series.
Incidentally, if you're curious why it converges to $\log(2)$ (wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 12,
"answer_id": 2
} |
Proof that $n^3+2n$ is divisible by $3$ I'm trying to freshen up for school in another month, and I'm struggling with the simplest of proofs!
Problem:
For any natural number $n , n^3 + 2n$ is divisible by $3.$
This makes sense
Proof:
Basis Step: If $n = 0,$ then $n^3 + 2n = 0^3 +$
$2 \times 0 = 0.$ So it is divisi... | We can take three cases (it's worth reading up on this by the way, the idea is called 'modular arithmetic)
$n≡0$ mod $3$
$n≡1$ mod $3$
$n≡2$ mod $3$
In the first case we are immmediately done because $n^3+2n=n(n^2+2)$ and since n factors out, by assumption we are done.
In the second and thirdcase we will substitute $'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 13,
"answer_id": 9
} |
Find the coordinates in an isosceles triangle Given:
$A = (0,0)$
$B = (0,-10)$
$AB = AC$
Using the angle between $AB$ and $AC$, how are the coordinates at C calculated?
| Let $a,b$ and $c$ be the side lengths and $A,B$ and $C$ the angles.
$a^{2}=x^{2}+\left( y+10\right) ^{2}$
$b^{2}=x^{2}+y^{2}=10^{2}$
$b=c=10$
By the (Neper) theorem of tangents (corollary of the Law of tangents):
$\tan \frac{A-B}{2}=\frac{a-b}{a+b}\cot \frac{C}{2}$
On the other hand
$\frac{A+B}{2}=\frac{\pi }{2}-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Finding the $N$-th derivative of $f(x)=\frac {x} {x^2-1}$ I'm practicing some problems from past exams and found this one:
Find the n-th derivative of this function:
$$f(x)=\frac {x} {x^2-1}$$
I have no idea how to start solving this problems. Is there any theorem for finding nth derivative?
| Maybe we can add some more help -just in case you didn't succeed to find the answer yourself yet. Let
$$
\frac{x}{x^2 - 1} = \frac{A}{x-1} + \frac{B}{x+1}
$$
be the splitting into partial fractions. (I'm too lazy to compute the coeffitients $A$ and $B$.) Then
$$
\frac{d}{dx} \frac{x}{x^2 - 1} = -\frac{A}{(x-1)^2} - \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
Simplification of expressions containing radicals As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$.
But we can also find the roots using Cardano's method, which leads to
$$x = \sqrt[3]{\sqrt{28/27} + 1} - \sqrt[3]{\sqrt{28/27} - 1}$$
and two other roots... | Yes. The first thing to try is to guess that $\sqrt[3]{ \left( \sqrt{ \frac{28}{27} } \pm 1 \right) } = \pm \frac{1}{2} + \sqrt{a}$ for some $a$. Cubing both sides then gives
$$\frac{2}{9} \sqrt{21} \pm 1 = \pm \frac{1}{8} + \frac{3}{4} \sqrt{a} \pm \frac{3}{2} a + a \sqrt{a}.$$
Setting $1 = \frac{1}{8} + \frac{3a}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
} |
The Basel problem As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it w... | Another proof i have (re?)discovered.
I want to prove that,
$\displaystyle J:=\int_0^1 \frac{\ln(1+x)}{x}dx=\frac{\pi^2}{12}$
Let $f$, be a function, such that, for $s\in[0;1]$,
$\displaystyle f(s)=\int_0^{\frac{\pi}{2}} \arctan\left(\frac{\sin t}{\cos t+s}\right)\,dt$
Observe that,
$\begin{align} f(0)&=\int_0^{\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "814",
"answer_count": 48,
"answer_id": 41
} |
Evaluate $\int \frac{1}{\sin x\cos x} dx $ Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However... | If I take the derivative of your second answer (call it $g(x)$), I get:
\begin{eqnarray*}
\frac{dg}{dx}
& = & -\frac{-\sin x}{\cos x} + \frac{\sin x}{2(1-\cos x)} + \frac{-\sin x}{2(1+\cos x)}\\
& = & \frac{\sin x\left(1-\cos^2 x + \frac{1}{2}\cos x(1+\cos x) - \frac{1}{2}\cos x(1-\cos x)\right)}{\cos x(1-\cos x)(1+\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 0
} |
function asymptotic where $f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$ If $a$ and $b$ are positive real numbers, and if $f(x)$ has the following asymptotic property
$f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$
then is the following true?
$f(x) = \frac{a}{b} + O(\frac{1}{\... | Yes. One way to see this is to actually do the long division (like the kind you learned in elementary school)! Unfortunately, typesetting that in full on this forum will overtax my LaTeX powers.
Anyway, dividing $b + O\left(\frac{1}{\sqrt{x}}\right)$ into $a + O\left(\frac{1}{\sqrt{x}}\right)$ yields $\frac{a}{b}$ wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Algebraic Identity $a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$ Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$.
So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial argument?
| Can we build a combinatorial argument along these lines?
Say if we have say $n$ students to be allotted in $a$ rooms with $b$ of the $a$ room being non air-conditioned. (Assume the students are distinguishable so we can order the student as $1,2,3,...,n$)
The total number of ways is $a^n$.
Suppose all students are allo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 2
} |
Limit of $S(n) = \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$ - Part II This is a follow up of Limit of $S(n) = \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$
More details can be found in the above thread.
Let $S(n) = \displaystyle \sum_{k=1}^{... | Here's an argument that pushes the lower bound for $S(n)$ closer to a factor of $2$ times
$\log_2 n.$ More precisely, we show
$$ S(n) \ge \left( 2 - \frac{1}{e} \right) \left( \lfloor \log_2 n \rfloor – 1 \right). $$
Let $ a= \lfloor \log_2 n \rfloor $ and write
$ f(n,x) = 1 - \prod_{j=1}^{n-1} ( 1 – jx),$
and so
$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Suggest a tricky method for this problem Find the value of:
$$ 1+ \biggl(\frac{1}{10}\biggr)^2 + \frac{1 \cdot 3}{1 \cdot 2} \biggl(\frac{1}{10}\biggr)^4 + \frac{1\cdot 3 \cdot 5}{1 \cdot 2 \cdot 3} \biggl(\frac{1}{10}\biggr)^6 + \cdots $$
| Note that we can rewrite the series as
$$ \displaystyle\sum\limits_{n=0}^\infty \frac{(2n)!}{2^n(n!)^2} \cdot \bigg(\frac{1}{10}\bigg)^{2n}$$
which is exactly
$$ \displaystyle\sum\limits_{n=0}^\infty \frac{(2n)!}{2^n(n!)^2} \cdot x^{2n}$$
evaluated at $x = \frac{1}{10}$. It is easy to see that this sum has radius of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/12302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
In every power of 3 the tens digit is an even number How to prove that in every power of $3$, with natural exponent, the tens digit
is an even number?
For example, $3 ^ 5 = 243$ and $4$ is even.
| A slightly different approach:
Let $n$ be the exponent of the variable power of $3$. Therefore, the power of $3$ takes the form $3^n$. Now, $n$ can either be even or odd. Hence two cases are formed:-
Case1: $n$ is even.
Since $n$ is even, let us say it is of the form $2k$.
Therefore, $3^n = 3^{2k} = (3^k)^2$. This mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/13890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 8,
"answer_id": 5
} |
Golden Number Theory The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers $\mathbb{Z}[\varphi]$ used in number theory though. If anyone happens to know some equations we can apply this in and h... | Formally the Riemann zeta-function can be expressed as
$$ \zeta(z)=\prod_{k=0}^{\infty}\;\;\prod_{p\in \mathbb{P}}\bigg\{\left( 1 -\varphi^{-1}\;p^{-5^{k}z} +p^{-2\cdot5^{k}z}\right)\left( 1 +\varphi\;p^{-5^{k}z} +p^{-2\cdot5^{k}z}\right)\bigg\} \;for\;z>1$$
where
$
\varphi=\frac{1+\sqrt{5}}{2}
$
is the Golden Ratio.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/18589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "61",
"answer_count": 4,
"answer_id": 2
} |
How to deduce trigonometric formulae like $2 \cos(\theta)^{2}=\cos(2\theta) +1$? Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)... | For Proving $\sin(\alpha+\beta)=\sin\alpha\cdot \cos\beta + \cos\alpha \cdot \sin\beta$ you can see this link:
*
*http://www.math.wisc.edu/~leili/teaching/math222s11/problems/quizzes/trig.pdf
By the above by substituting $\beta=\alpha$ you have
*
*$\sin{2\alpha} = \sin\alpha \cdot \cos\alpha + \cos\alpha \cdot \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 2
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.