Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How to know that $a^3+b^3 = (a+b)(a^2-ab+b^2)$ Is there a way of go from $a^3+b^3$ to $(a+b)(a^2-ab+b^2)$ other than know the property by heart?
| So I guess the answer to this question is to expand the right-hand side of the equation, although it might not be too clear on how to do this.
It might be clearer if you make the substitution $u=a+b$. Then the right hand side is:
\begin{align*}
(a+b)(a^2-ab+b^2) & = u(a^2-ab+b^2) \\
& = ua^2-uab+ub^2 \\
& = (a+b)a^2-(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/20301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 2
} |
Simple limit, wolframalpha doesn't agree, what's wrong? (Just the sign of the answer that's off) $\begin{align*}
\lim_{x\to 0}\frac{\frac{1}{\sqrt{4+x}}-\frac{1}{2}}{x}
&=\lim_{x\to 0}\frac{\frac{2}{2\sqrt{4+x}}-\frac{\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\lim_{x\to 0}\frac{\frac{2-\sqrt{4+x}}{2\sqrt{4+x}}}{x}\\
&=\... | In between the fourth and fifth steps, you go from
$$\lim_{x\to 0}\frac{2-\sqrt{4+x}}{2x\sqrt{4+x}} \text{ to } \lim_{x\to 0}\frac{(2-\sqrt{4-x})(2+\sqrt{4-x})}{(2x\sqrt{4+x})(2+\sqrt{4-x})}$$
which is not correct. It should be
$$\lim_{x\to 0}\frac{(2-\sqrt{4+x})(2+\sqrt{4+x})}{(2x\sqrt{4+x})(2+\sqrt{4+x})}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/22704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Prime factorization of square numbers Let n be a natural number with unique prime factorization $p^m$... $q^k$ . Show that n can be written as a square if and only if all (m, ...k) are even
| If $$n = \prod_{i = 1}^\infty {p_i}^{\alpha_i} \in \mathbb Z,$$ where $p_i$ are the primes in order by $i$ and $\alpha_i$ are the corresponding exponents which may be $0$ as needed, we then have $$\sqrt n = \sqrt{\prod_{i = 1}^\infty {p_i}^{\alpha_i}} = \prod_{i = 1}^\infty {p_i}^{\frac{\alpha_i}{2}}.$$ But if any of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/23360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Why determinant of a 2 by 2 matrix is the area of a parallelogram? Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$.
How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$?
Are the areas of the following parallelograms the same?
$(1)$ parallelogram with ver... | For the matrix $\left[\begin{array}{cc}
a & c \\
b & d \\
\end{array}\right]$ let
$$A = \left[\begin{array}{c}
a \\
b \\
\end{array}\right]
\;\text{and}\; B = \left[\begin{array}{c}
c \\
d \\
\end{array}\right]$$
as shown in the following figure.
Then the height of the parallelogram is
$$\text{height} = |B|\sin\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/29128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "80",
"answer_count": 11,
"answer_id": 5
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How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that ther... | I first encountered this sum with the following problem:
Evaluate
$$\bigg(\frac{1}{2}\bigg)^\dfrac{1}{3}\bigg(\frac{1}{4}\bigg)^\dfrac{1}{9}\bigg(\frac{1}{8}\bigg)^\dfrac{1}{27}\bigg(\frac{1}{16}\bigg)^\dfrac{1}{81}\dots$$
Which , of course simplified to
$$\bigg(\frac{1}{2}\bigg)^{\dfrac{1}{3^1}+\dfrac{2}{3^2}+\dfra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "438",
"answer_count": 23,
"answer_id": 8
} |
All natural solutions of $2x^2-1=y^{15}$ How can I find all positive integers $x$ and $y$ such that $2x^2-1=y^{15}$?
PS. See here.
| $(1,1)$ is available by inspection. Otherwise, if you write $2x^2=y^{15}+1=(y+1)(y^2-y+1)(y^4-y^3+y^2-y+1)(y^8+y^7-y^5-y^4-y^3+y+1)$ (Thanks, Alpha) you can look at where the factors of $2x^2$ can come from. As the last three terms are odd, $y+1$ must have an odd numbers of factors of $2$. $y^4-y^3+y^2-y+1=(y^3-2y^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Complete induction of $10^n \equiv (-1)^n \pmod{11}$ To prove $10^n \equiv (-1)^n\pmod{11}$, $n\geq 0$, I started an induction.
It's $$11|((-1)^n - 10^n) \Longrightarrow (-1)^n -10^n = k*11,\quad k \in \mathbb{Z}. $$
For $n = 0$:
$$ (-1)^0 - (10)^0 = 0*11 $$
$n\Rightarrow n+1$
$$\begin{align*}
(-1) ^{n+1} - (10) ^{n+... | If $a \equiv b \left(\bmod m \right)$, then $a^n \equiv b^n \left(\bmod m \right)$ where $n \in \mathbb{N}$ and $a,b,m \in \mathbb{Z}$.
The reason being $a^n - b^n$ factors as $(a-b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + b^{n-1})$ and hence if $m | (a-b)$, $m | (a^n-b^n)$
If you want to prove by induction, first n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/39882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 12,
"answer_id": 3
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Primitive polynomials of finite fields there are two primitive polynomials which I can use to construct $GF(2^3)=GF(8)$:
$p_1(x) = x^3+x+1$
$p_2(x) = x^3+x^2+1$
$GF(8)$ created with $p_1(x)$:
0
1
$\alpha$
$\alpha^2$
$\alpha^3 = \alpha + 1$
$\alpha^4 = \alpha^3 \cdot \alpha=(\alpha+1) \cdot \alpha=\alpha^2+\alpha$
$\alp... | The situation is not so different in a simpler context, the field of 5 elements, also known as the integers modulo 5. Whether $\alpha$ is $2$ or $3$, the field is $0,1,\alpha,\alpha^2,\alpha^3$, but whether $\alpha+\alpha+1=0$ depends on which $\alpha$ you choose.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/40326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Compound angle formula confusion I'm working through my book, on the section about compound angle formulae. I've been made aware of the identity $\sin(A + B) \equiv \sin A\cos B + \cos A\sin B$. Next task was to replace B with -B to show $\sin(A - B) \equiv \sin A\cos B - \cos A \sin B$ which was fairly easy. I'm strug... | Update: Since $\sin (\frac{\pi }{2}-x)\equiv \cos x$, you have
$$\sin \left(\frac{\pi }{2}-A-B\right)\equiv \sin \left(\frac{\pi }{2}-(A+B)\right)\equiv \cos (A+B).$$
Replacing $A$ by $\frac{\pi }{2}-A$ in
$$\sin (A-B)\equiv \sin A\cos B-\cos A\sin B,$$
gives
$$\sin \left(\frac{\pi }{2}-A-B\right)\equiv \sin \left( \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Constructing idempotent matrices Is there a general method for constructing an idempotent matrix if we are given the values of the diagonal entries?
| The following should work "generically" (given a list of diagonal elements $d_i, i=1\ldots n$ where $\sum_{i=1}^n d_i = r$ is an integer, $1 \le r \le n-1$). Start with a diagonal matrix with $r$ $1$'s and $n-r$ $0$'s on the diagonal. First conjugate with a random matrix.
Then for $k$ from 1 to $n-1$, conjugate with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$ How do I show that:
$$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$
This is actually problem B $4371$ given at this link. Looks like a... | This may be a $3$ year old question, but I would like to add to the list an answer that relies on the sum of $\tan^2$ identity.
Let $$\begin{align}S &= \frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}}\\
&=\frac{1}{\cos^{2}\frac{3\pi}{7}} + \frac{1}{\sin^{2}\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 5,
"answer_id": 2
} |
linear algebra - equations
Determine real number(s)for $a,b$ such that the system has no solution, has a unique solution, and has more than one solution:
$$\begin{align*}
x-2y+az-t&=1\\
-x+y-z+t&=-1\\
(a+1)y-a^2z+at&=0\\
(b+1)y-abz-a^2t&=b
\end{align*}$$
I could not transform the matrix into row reduced form... | Your matrix will have $a$ and $b$ indicated. You then have to be careful with them.
(I hope I didn't make any silly arithmetic or algebra mistakes; but even if I did, you should be able to see how one proceeds: doesn't matter what $a$ and $b$ are, so long as you are careful not to divide by an expression containing th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Exponents in the denominator? I'm having trouble understanding exponents in the denominator.
For example: I have the expression: $\displaystyle 1 - \frac{1}{3^n} + \frac{2}{3^{n+1}}$.
I know that this simplifies to $\displaystyle 1 - \frac{1}{3^{n+1}}$, but how/why? Can someone please list the steps?
My understanding i... |
My understanding is that the exponent $n+1$ means that $x^{n+1}=x\cdot x^n$
Your understanding is correct, and you can apply it to the problem at hand by remembering that when an exponent, say $n$, of an expression is in the denominator of a fraction, it can be written as an expression raised to the $-n$ power. Then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/47180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Deriving the rest of trigonometric identities from the formulas for $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, and $\cos (A-B)$ I am trying to study for a test and the teacher suggest we memorize $\sin(A+B)$, $\sin(A-B)$, $\cos(A+B)$, $\cos (A-B)$, and then be able to derive the rest out of those. I have no idea how to ge... | I'm not certain, but I think perhaps you need to revisit the basic trigonometric definitions.
$$\begin{array}{lll}
\sin\theta = \frac{opposite}{hypotenuse}&\csc\theta=\frac{hypotenuse}{opposite}\\
\cos\theta = \frac{adjacent}{hypotenuse}&\sec\theta=\frac{hypotenuse}{adjacent}\\
\tan\theta = \frac{opposite}{adjacent}&\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
Why can ALL quadratic equations be solved by the quadratic formula? In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use it. I have tried to figure it out by proving these t... | I'll just let the algebra speak for itself.
For $ a \ \ne \ 0 $
If
$ax^2 = bx + c $
Then
$ x = \frac{b \ \pm \sqrt{b^2 \ + \ 4ac}}{2a} $
PROOF:
$ 4aax^2 = 4abx + 4ac $
$(2ax)^2 + b^2 = 4abx + 4ac + b^2 $
$(2ax)^2 - 4abx + b^2 = b^2 + 4ac $
$ (2ax - b)^2 = b^2 + 4ac $
$ 2ax - b = \pm \sqrt{b^2 + 4ac} $
$2ax = b \pm \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/49229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "294",
"answer_count": 22,
"answer_id": 2
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Property of an ellipse I need proof for the following question. Also, I want to know, can we apply the same for other conics. If yes, where and when... Please explain.
Show that there exists a point K on the major axis of E , having the property that for any chord $\overline{PQ}$ passing through K, $\dfrac{1}{PK^2} + ... | We use a brute force approach. Let our ellipse have equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
where $a \ge b >0$.
Let $K=(p,0)$ be a point (if there is one) such that $1/(PK)^2+1/(QK)^2$ is the same for all chords $PQ$ through $K$.
We will, regrettably, cheat, by (sort of) looking up the answer. Suppose that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/52359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How $x^2 + 5x + 6 \ge 0$ implies $−3 \ge x \ge −2$? How $x^2 + 5x + 6 \ge 0$ implies $−3 \ge x \ge −2$ ?
I represented $x^2 + 5x + 6 \ge 0$ as $(x+3)(x+2) \ge 0$,which means that either $(x+3) \ge 0$ and $(x+5) \ge 0$ which giving $x \ge -3$ or $x \ge -2$ but how does this gives $−3 \ge x \ge −2$?
ADDED: The above expr... | $x^{2}+5x+6=0$, Solutions are $x=-3,x=-2.$
Since $x^{2}+5x+6=(x+3)(x+2)$, we have three cases:
*
*If $x\geq -2$, then $x+3\geq 1$, $x+2\geq 0$, so $x^{2}+5x+6\geq 0.$
*If $x\leq -3$, then $x+3\leq 0$, $x+2\leq -1$, so $x^{2}+5x+6\geq 0.$
*If $-3<x<-2$, then $x+3>0$, $x+2<0$, so $x^{2}+5x+6<0.$
Remark: instead of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/53396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Construction of generating function from identity I am trying to solve identity involving binomials and Fibonacci numbers by using generating functions:
$$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose k}(-1)^{n-k}f_{2k+1}$$
My computational approach by Mathematica lead me to derive... | Here is a proof using complex variables. We seek to show that
$$\sum_{k=0}^n {n\choose k} {n+k\choose k} F_{k+1}
=\sum_{k=0}^n {n\choose k} {n+k\choose k} (-1)^{n-k} F_{2k+1}.$$
Start from
$${n+k\choose k}
= \frac{1}{2\pi i}
\int_{|z|=1} \frac{1}{z^{k+1}} (1+z)^{n+k} \; dz.$$
This yields the following expression for t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/53830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
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Is this Batman equation for real? HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{... | Here's the equations typed out if you want save time with writing it yourself.
(x/7)^2*SQRT(ABS(ABS(x)-3)/(ABS(x)-3))+(y/3)^2\*SQRT(ABS(y+3*SQRT(33)/7)/(y+3*SQRT(33)/7))-1=0
ABS(x/2)-((3*SQRT(33)-7)/112)*x^2-3+SQRT(1-(ABS(ABS(x)-2)-1)^2)-y=0
9*SQRT(ABS((ABS(x)-1)*(ABS(x)-0.75))/((1-ABS(x))*(ABS(x)-0.75)))-8*ABS(x)-y=0
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "466",
"answer_count": 10,
"answer_id": 8
} |
Rule for multiplying rational numbers The rule for multiplying rational numbers is this:
$\space\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$
Can the rule be proven or is it meant to be taken as a given?
Edit: Where $b\neq 0$ and $d\neq 0$.
| An approach to make it more clear might be to seperate each rational into a product of its parts ie. $\frac {a}{b}= \frac{a}{1} \cdot \frac{1}{b}$ and $\frac {c}{d}= \frac{c}{1} \cdot \frac{1}{d}$ then use the commutative property to group the "numerator" fractions and the "denominator" fractions seperately: $\frac{a}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/58174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
"answer_id": 0
} |
How many triplets of real numbers $(x, y, z)$ which satisfy these $3$ restriction: How many triplets of real numbers $(x, y, z)$ which satisfy :
$$(x + y)^3 = z$$
$$(y + z)^3 = x$$
$$(z + x)^3 = y$$
I need some approaches for solving this problem.
| Look first for solutions of the shape $x=y=z=a$. Our equations all reduce to
$(2a)^3=a$, which has the solutions $a=0$ and $a=\pm 2^{-3/2}$.
Now look for solutions where not all the variables are equal. For definiteness, look for solutions with $x \lt z$.
If $x \lt z$, then $x+y \lt y+z$ and therefore $(x+y)^3 \lt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/60147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
| $$n^3=S_n-S_{n-1}=\left(\frac{n(n+1)}2\right)^2-\left(\frac{n(n-1)}2\right)^2=n^2\left(\frac{n+1}2-\frac{n-1}2\right)\left(\frac{n+1}2+\frac{n-1}2\right)\\
=n^2\cdot1\cdot n.$$
This is the inductive step. The rest is easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/62171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
"answer_count": 16,
"answer_id": 4
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Is $\frac{1}{(\textrm{can}-a-b\cdot i)^2}-\frac{1}{(\textrm{can}+\textrm{cod}-a-b \cdot i)^2}$ in any way simplifiable with Maple? So I wonder if Maple can reduce such lines like $$\frac{1}{(\textrm{can}-a-b\cdot i)^2}-\frac{1}{(\textrm{can}+\textrm{cod}-a-b \cdot i)^2}$$ (assuming all variables but $i$ are real)?
| Writing $x = can - a - i\cdot b$ and $y = cod$, your expression is equivalent to
$$\frac{1}{x^2} - \frac{1}{(x + y)^2}$$
You could try reducing it to a single fraction:
$$\frac{1}{x^2} - \frac{1}{(x + y)^2} = \frac{(x + y)^2 - x^2}{x^2 (x + y)^2} = \frac{2xy + y^2}{x^2 (x + y)^2} = \frac{y(2x + y)}{x^2 (x + y)^2}$$
I g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/63414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Product of $ |z^k - 1| $ Problem: Prove the following identity about the product involving the nth roots of unity:
$$
\prod_{k=1}^{N-1}|z^k-1| = N
$$
where $ z^k $ is the primitive nth root of unity.
Attempt:
$$
\begin{align}
\prod_{k=1}^{N-1}|z^k-1| &= \prod_{k=1}^{N-1}\left|(\cos(\frac{2\pi k}{N})-1)+i\sin(\frac... | Aha, I just solved it:
First consider the polynomial $$ \prod_{k=0}^{N-1}(x-z^k) $$
The roots of the polynomial are the nth roots of unity, which are precisely the roots of the polynomial $ x^n-1 $ and so the two are equal.
Dividing both sides by $ x-1 $, we get
$$ \prod_{k=1}^{N-1}(x-z^k) = 1+x+\dots+x^{N-1} $$
Substi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/65744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
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Find x to keep the equality $\sqrt[2x+1]{\sqrt[11-4x]{(-2x)^{3x}}}=\sqrt[3x-1]{7x+2}$ $$\mbox{ Find }x \in \mathbb{Q} \mbox{ to keep the equality: } \sqrt[2x+1]{\sqrt[11-4x]{(-2x)^{3x}}}=\sqrt[3x-1]{7x+2}$$
I tried to write the roots using powers:
\begin{align*}\sqrt[2x+1]{\sqrt[11-4x]{(-2x)^{3x}}}=\sqrt[3x-1]{7x+2}&\R... | I agree with Peter Taylor.
You get:
\begin{equation}
\frac{3x}{(2x+1)(11-4x)}\ln(-2x) = \frac{1}{3x-1}\ln(7x+2)
\end{equation}
Graphing gives about -0.15649: http://bit.ly/q5Rn4l
I don't think you can solve it exactly. A rational solution is possible but unlikely in general. Maybe you're meant to just try a whole bunch... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Show that $n + 2$ and $n^2 + n + 1$ cannot both be perfect cubes
Question: If $n$ is a nonnegative integer, prove that $n + 2$ and $n^2 + n + 1$ cannot both be perfect cubes.
Possible solution: Suppose $n+2$ and $n^2 + n + 1$ are perfect cubes, their product $(n+2)(n^2 + n + 1)$ must also be a perfect cube.
However, n... | Except for $-1$, $0$ and $1$, the distance between consecutive perfect cubes is always greater than one. This is enough to conclude that $(n+1)^3+1$ is not a perfect cube when $n$ is nonnegative. (No need to invoke Fermat.)
| {
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"url": "https://math.stackexchange.com/questions/71412",
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"source": "stackexchange",
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Why is the math for negative exponents so? This is what we are taught:
$$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$
but I don't understand why we take the inverse of the base when we have a negative exponent. Can anyone explain why?
| $$
\begin{align}
a\cdot 3^{10} & = a\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 \\ \\
a\cdot 3^6 & = a\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3
\end{align}
$$
To go from the second line to the first, multiply by 3 four times.
To go from the first to the second, multiply by 3 minus four... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Where do I fail with this integral (coordinates on a simplex)? Let us regard the standard simplex in two dimensions
$S = \{ x_1 + x_2 \leq 1, x_1 \geq 0, x_2 \geq 0 \}$
Where does the following calculation fail:
$ \int_S x_1^2 = \int_0^1 x_1^2 \int_0^{1-x_1} dx_2 dx_1
= \int x_1^2 ( 1 - x_1 ) dx_1 = \int_0^1 x_1^2 - x... | There is an oversight in the second calculation.
You wanted
$$\int \frac{(1-x_2)^3}{3}\,dx_2,$$
and decided to expand $(1-x_2)^3$ and integrate term by term.
Note that by the Binomial Theorem, or otherwise,
$$(1+t)^3=1+3t+3t^2+t^3,$$
so $(1-x_2)^3=1-3x_2 +3x_2^2-x_2^3$.
Expanding is in any case an inefficient way to e... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which Möbius transformations send 0, 1 and infinity to 0, 1 and infinity
Possible Duplicate:
How do I find out the symmetry of a function?
Let $f:\mathbf{P}^1 \longrightarrow \mathbf{P}^1$ be a Möbius transformation $z\mapsto (az+b)/(cz+d)$ sending $\{0,1,\infty\}$ to $\{0,1,\infty\}$ with $ad-bc = 1$.
I suspect th... | If $0\mapsto 0$, then $b=0$. This means that $d=\frac{1}{a}$.
*
*If $f(1)=1$ and $f(\infty)=\infty$, then we must have $a=c+d$ and $c=0$, so $ad=1$, $a=1$. Hence $a=d=\pm 1$, $b=c=0$. The only transformation is the identity.
*If $f(1)=\infty$ and $f(\infty)=1$, then we must have $a=c$ and $c+d=0$; since $d=\frac{1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement
$$\lim_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out... | Let $\sin(x)$ is defined as solution of $\frac{d^2}{dx^2}\textrm{f}(x)=-\textrm{f}(x)$ with $\mathrm f(0)=0,\,\frac{d}{dx}\mathrm f(0)=C$ initial conditions, so exact solution is $\mathrm f(x)=C\cdot\sin(x)$.
Define second derivative as
$$
\begin{align*}
\frac{d^2}{dx^2}\textrm{f}(x)=\lim_{\Delta x\to 0}{\frac{\frac{... | {
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"url": "https://math.stackexchange.com/questions/75130",
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"source": "stackexchange",
"question_score": "536",
"answer_count": 28,
"answer_id": 20
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Interscholastic Mathematic League Senior B Division #11 The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
| We solve the problem using too much machinery for a contest question that has to be solved quickly!
The following result is useful. Let $r_1$, $r_2$, and $r_3$ be the roots of the cubic equation $x^3+Ax^2+Bx+C=0$. Then (i) $r_1+r_2+r_3=-A$; (ii) $r_1r_2+_2r_3+r_3r_1=B$; and (iii) $r_1r_2r_3=-C$.
The roots of our cubic... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all odd primes $p$ for which $15$ is a quadratic residue modulo $p$ I want to find all odd primes $p$ for which $15$ is a quadratic residue modulo $p$.
My thoughts so far: I want to find $p$ such that $ \left( \frac{15}{p} \right) = 1$. By multiplicativity of the Legendre symbol, this is equivalent to $ \left( \fr... | You must also consider your primes mod 4, as that alters the behavior of reciprocity. So, in the order I found them, all values are
$$ 1,49; \; 17, 53; \; 11, 59; \; 7, 43 \pmod {60}.$$ In ordinary numerical order,
$$1, 7,11,17,43,49, 53,59 \pmod {60}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/77351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 0
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How could we find the largest number in the sequence $ \sqrt{50},2\sqrt{49},3\sqrt{48},\cdots 49\sqrt{2},50$? How to find the largest number in the sequence$$ \sqrt{50},2\sqrt{49},3\sqrt{48},\cdots 49\sqrt{2},50$$
I am interested in a "calculus-free" approach.
Thanks,
| The $n$-th term in the sequence is $n\sqrt{51-n}=\sqrt{n^2(51-n)}$. So the question is: for which $n$ ($1\le n\le 50$), does $n^2(51-n)$ become the largest?
If you want to avoid calculus, you could use the AM-GM inequality:
if $x,\,y,\,z\ge 0$, then $$\frac{x+y+z}{3}\ge\sqrt[3]{xyz},$$
with equality if and only if $x=y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
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How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$? How can one prove that $\sqrt[3]{\left ( \frac{a^4+b^4}{a+b} \right )^{a+b}} \geq a^ab^b$, $a,b\in\mathbb{N^{*}}$?
| Here is a much more elementary proof:
$$a^{3a}b^{3b}=a^3a^3 \cdot... a^3 b^3b^3 \cdot ....b^3 \,.$$
Using the AM-GM inequality with $x_1=...=x_a=a^3$ and $x_{a+1}=...=x_{a+b}=b$ Yields
$$\sqrt[a+b]{a^3a^3 \cdot ... a^3 b^3b^3 \cdot ....b^3} \leq \frac{aa^3+bb^3}{a+b} \,.$$
Thus
$$a^{3a}b^{3b} \leq \left( \frac{a^4+b^4}... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $f(x)$ is a polynomial satisfying $ f(x)f(\frac 1x) = f(x)+f(\frac 1x)$ and $f(3)=28$, then how could we find $f(4)$?
If $f(x)$ is a polynomial satisfying $ f(x)f(\frac 1x) = f(x)+f(\frac
1x)$ and $f(3)=28$, then how could we find $f(4)$ ?
| Solving the functional equation for $f\left(\frac{1}{x} \right) = \frac{f(x)}{f(x)-1}$. This means that $f(x)-1$ must be a monomial. Let $f(x) = 1 + c x^d$. Then
$$
c \left( \frac{1}{x} \right)^d +1 = \frac{1}{c} \left( \left( \frac{1}{x} \right)^d + c \right)
$$
This, implies $c^2 = 1$. Now use $f(3) = 28$ to de... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Continued fraction: Show $\sqrt{n^2+2n}=[n; \overline{1,2n}]$ I have to show the following identity ($n \in \mathbb{N}$):
$$\sqrt{n^2+2n}=[n; \overline{1,2n}]$$
I had a look about the procedure for $\sqrt{n}$ on Wiki, but I don't know how to transform it to $\sqrt{n^2-2n}$.
Any help is appreciated.
EDIT:
I tried the fo... | Because $n^2+2n=(n+1)^2-1$, the leading term in the continued fraction expansion will be $n$. Multiplying by a form of $1$, we have
$$\sqrt{n^2+2n}-n=\frac{2n}{n+\sqrt{n^2+2n}}$$
and $1<\frac{n+\sqrt{n^2+2n}}{2n}<2,$ so the next term will be $1$. Since $\frac{n+\sqrt{n^2+2n}}{2n}-1=\frac{\sqrt{n^2+2n}-n}{2n}$ and we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Discriminant of derivative of cubic equation being a perfect square Is it possible for the discriminant of the first derivative of a cubic polynomial $(x+a)(x+b)(x+c)$, where $a, b$ and $c$ are distinct non-zero integers (i.e. Discriminant $[d((x+a)(x+b)(x+c))/dx]$ in Wolfram|Alpha/Mathematica) to be a perfect square?
... | Let $p(x) = x^3 + (a + b + c)x^2 + ((b + c)a + bc)x + abc$. Then,
$$p'(x) = 3x^2 + 2(a+b+c)x + ((b+c)a+bc).$$
The discriminant of $p'(x)$ is
$$\Delta = 4(a+b+c)^2 - 4\cdot 3\cdot ((b+c)a+bc)=4(a^2 - (b + c)a + (b^2 - bc + c^2)).$$
So you need to find values of $a,b,c$ that make $\Delta$ a square... Notice that of cour... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Solving quadratic equation $$\frac{1}{x^2} - 1 = \frac{1}{x} -1$$
Rearranging it I get: $1-x^2=x-x^2$, and so $x=1$. But the question Im doing says to find 2 solutions. How would I find the 2nd solution?
Thanks.
| $$\frac{1}{x^2} - 1 = \frac{1}{x} -1$$
$$\frac{1}{x^2} = \frac{1}{x} $$
multiply by x suppose that $x \ne0$
$$x^2-x=0$$
$$x(x-1)=0$$
The unique solution is $x=1$ solution $x=0$ is excluded.
| {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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"answer_id": 5
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What is the result of $\lim\limits_{x \to 0}(1/x - 1/\sin x)$? Find the limit:
$$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)$$
I am not able to find it because I don't know how to prove or disprove $0$ is the answer.
| METHOD I
Firstly, notice that the expression under the limit is an odd function and consider that $\sin(x)<x$. Then we have that:
$$\lim_{x \rightarrow 0}\left(\frac1x - \frac1{\sin x}\right)= \lim_{x \rightarrow 0}\frac{\sin x - x}{x\sin x}\le\lim_{x \rightarrow 0}\frac{\sin x - x}{x^2}\le\lim_{x \rightarrow 0}\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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A Limit problem : No-existence I find difficulty proving the no existence of this limit
I show my process
$$ \lim_{x\to 0} \biggl(1 + x e^{- \frac{1}{x^2}}+\sin \frac{1}{x^4}\biggr)^{e^{\frac{1}{x^2}}}$$
We begin with rewriting the limit as follows:
$$ \lim_{x\to 0} \biggl(1 + x e^{- \frac{1}{x^2}}+\sin \frac{1}{x^4}... | Given an integer $k$, let $\frac{1}{x^4}=2\pi k + \frac{\pi}{2}$, or $x = (2\pi k + \frac{\pi}{2})^{-\frac{1}{4}}$. Then $\sin {\frac{1}{x^4}} = 1$. Let $y=e^{\frac{1}{x^2}}$. Then your expression is:
$$(2+\frac{x}{y})^y$$.
Now, if $|x|<1$ then $y>e$, so $\frac{x}{y}> \frac{-1}{2}$. So this expression is at least as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do we calculate the area of a region bounded by four different curves? Calculate the area(express both respectively in integral with one variable) bounded by the following curves (i.e. the shape with one side corresponding to one curve):
$$xy=1, \quad xy^2=3,\quad x^2-y^2=26,\quad x^2-y^3=11$$
This problem is creat... | As long as you're only asking for an expression as an integral, and not an actual number, we can calculate the area as follows:
Let
*
*$a$ be the positive real solution of $x^5-11x^3-1=0$
*$b$ be the positive real solution of $x^{7/2}-11x^{3/2}-3\sqrt{3}=0$
*$c$ be the positive real solution of $x^4-26x^2-1=0$... | {
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"timestamp": "2023-03-29T00:00:00",
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difficult sequence Prove that the sequence:
$$a_n= \frac{1}{n}\left(e\cdot\sqrt e \cdots\sqrt[3]e\cdot\sqrt[n]e\right)$$
is decreasing to a finite limit. After having shown that the sequence:
$$b_n=\left(\sum_{k=1}^n\frac{1}{k}\right)-\log n$$
converges to a positive real number $b,$ say who is the limit of $ a_n $
| $$\eqalign{
& {a_n} = \frac{1}{n}\prod\limits_{k = 1}^n {{e^{\frac{1}{k}}}} \cr
& \log {a_n} = - \log n + \sum\limits_{k = 1}^n {\frac{1}{k}} \cr
& \log {a_n} = \sum\limits_{k = 1}^n {\frac{1}{k}} - \log n \cr
& \mathop {\lim }\limits_{n \to \infty } \log {a_n} = \gamma \cr} $$
You can prove $0 < \ga... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Uniqueness of polynomial Let $n$ be a nonnegative integer. Can you help me prove the following ?
There exists a unique polynomial $P_{n}$ such that for all $t \in [0,\frac{\pi}{2}]$,
$P_{n}(\operatorname{cotan}^2t)=\frac{\sin((2n+1)t}{(\sin t)^{2n+1}}$
with $\operatorname{cotan}(x)=\frac{\cos x}{\sin x}$.
| There may be an easier way but this is what I came up with:
Start with Euler's formula:
\begin{align}
e^{i(2n+1)t} &= \cos((2n+1)t) + i \sin((2n+1)t) = (\cos t + i \sin t)^{2n+1} \\
&= \sum_{k=0}^{2n+1} \binom{2n+1}{k}i^k \sin^k t \cos^{2n+1-k}t \\
\end{align}
Now we separate the sum for even and odd valu... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Finding the limit of $\frac{1}{t\sqrt{1+t}} - \frac{1}{t}$ as $t$ tends to $0$ $$\lim_{t\rightarrow 0}\left(\frac{1}{t\sqrt{1+t}} - \frac{1}{t}\right)$$
I attemped to combine the two fraction and multiply by the conjugate and I ended up with:
$$\frac{t^2-t^2\sqrt{1+t}}{t^3+{t\sqrt{1+t}({t\sqrt1+t})}}$$
I couldn't reall... | Perhaps you were trying something like
$\dfrac{1}{t\sqrt{1+t}} - \dfrac{1}{t} = \dfrac{1-\sqrt{1+t}}{t\sqrt{1+t}} = \dfrac{1-(1+t)}{t\sqrt{1+t}(1+\sqrt{1+t})} = \dfrac{-1}{\sqrt{1+t}(1+\sqrt{1+t})} $
which has a limit of $\dfrac{-1}{1 \times (1+1)} = -\dfrac{1}{2}$ as $t$ tends to $0$.
Added: If you are unhappy with t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
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Solutions of some Diophantine equations Respected Mathematicians,
The Diophantine equation
$$2^x + 5^y = z^2$$
has solutions $$x = 3, y = 0, z = 3$$ and $$x = 2, y = 1, z = 3$$ I got these solutions by trial and error method. To be honest, these solutions are below the number $5$. So, I easily verified them by trial ... | Assuming x, y, z are non-negative integers, there are no solutions to $4^x + 7^y = z^2$ and $4^x + 11^y = z^2$. This can be shown through modular arithmetic.
Taking the equation a) mod 3 you'll find the following:
*
*$4^x \equiv 1 \pmod{3}$
*$7^y \equiv 1 \pmod{3}$
*$z^2 \equiv 0$ or $1 \pmod{3}$
Since $1 + 1 = ... | {
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"url": "https://math.stackexchange.com/questions/100709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
evaluate $\int \frac{x^2\cdot\log(x)}{x+1} dx$ I am trying to evaluate integral:
$$\int \frac{x^2\log(x)}{x+1}dx$$
But I have some problems with it. If I use Wolfram Alpha like this
I get a result, but I need evaluate it by hand. Which method should I use?
If we represent it as:
$$\int(x^2\log(x)\cdot d(\ln(x))d... | We have
\begin{align*}
\int\frac{x^2\log x}{x+1}dx&=\int x\log x\frac{x+1-1}{x+1}dx\\
&=\int x\log x-\int \frac{x+1-1}{x+1}\log xdx\\
&=\frac{x^2}2\log x-\int\frac{x^2}2\frac 1xdx-\int \log x+\int\frac{\log x}{x+1}dx\\
&=\frac{x^2}2\log x-\frac{x^2}4-x\log x+x+\log x\log(x+1)-\int\frac{\log(x+1)}xdx\\
&=\frac x4(4-x)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Compute: $\int_{0}^{1}\frac{x^4+1}{x^6+1} dx$ I'm trying to compute: $$\int_{0}^{1}\frac{x^4+1}{x^6+1}dx.$$
I tried to change $x^4$ into $t^2$ or $t$, but it didn't work for me.
Any suggestions?
Thanks!
| one way is partial fractions on
$$
\frac{x^4+1}{x^6+1}=\frac{(x-e^{\pi i/4})(x-e^{3\pi i/4})(x-e^{5\pi i/4})(x-e^{7\pi i/4})}{(x-e^{\pi i/6})(x-e^{3\pi i/6})(x-e^{5\pi i/6})(x-e^{7\pi i/6})(x-e^{9\pi i/6})(x-e^{11\pi i/6})}
$$
$$
=\frac{(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)}{(x^2+1)(x^2+\sqrt{3}x+1)(x^2-\sqrt{3}x+1)}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 4
} |
Pythagorean triplets Respected Mathematicians,
For Pythagorean triplets $(a,b,c)$, if $c$ is odd then any one of $a$ and $b$ is odd. Here $(a, b, c)$ is a Pythagorean triplet with $c^2 = a^2 + b^2$.
Now, I will consider $c = b + \Omega$. The reason for considering $c = b + \Omega$ is, $c$ is a hypotenuse side of righ... | It is unclear to me what you are going for; for one thing, not every even number is a power of $2$. For another, even if $a$ is a power of $2$, you have no warrant to assert that $\Omega$ will necessarily be a power of $2$ as well; at least, no warrant that you have given. Of course, you may assume that's the case, but... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Prove $\frac{\cos^3{x}-\sin^3{x}}{\cos{x}-\sin{x}} =1+\frac{1}{2} \sin{2x}$ Prove
$$\frac{\cos^3{x}-\sin^3{x}}{\cos{x}-\sin{x}} =1+\frac{1}{2} \sin{2x}$$
How do I start :( which identity do I use?
| Hint: use the identity:
$(a^3-b^3)=(a-b)(a^2+ab+b^2)$.
Solution follows:
We have
$\cos^3 x -\sin^3 x =(\cos x-\sin x)(\cos^2 x +\cos x\sin x +\sin^2 x )$.
So:
$$\eqalign{{
\cos^3 x -\sin^3 x\over \cos x-\sin x}&=
{(\cos x-\sin x)(\cos^2 x +\cos x\sin x +\sin^2 x )\over (\cos x-\sin x)}\cr
&=1+\cos x\sin x\cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integral without using Euler substitution Help me please with integral:
$$\int \frac{2x-\sqrt{4x^{2}-x+1}}{x-1}\;dx$$
I must solve it without using Euler substitution.
Thanks!
| Hint: writing
$$\frac{2x-\sqrt{4x^2-x+1}}{x-1}=\frac{2x-\sqrt{4x^2-x+1}}{x-1}\frac{2x+\sqrt{4x^2-x+1}}{2x+\sqrt{4x^2-x+1}},$$
we find $$\frac{2x-\sqrt{4x^2-x+1}}{x-1}=\frac{4x^2-(4x^2-x+1)}{(x-1)(2x+\sqrt{4x^2-x+1})}=\frac 1{2x+\sqrt{4x^2-x+1}}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/103227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to define the sign of a function $$y=\arctan\frac{x+1}{x-3} + \frac{x}{4}$$
I know that is necessary to put the function $>$ than $0$, but then?
$$\arctan\frac{x+1}{x-3} + \frac{x}{4}>0$$
It's a sum, so I can't set up a "false system" putting the two factors $>0$. In this case which is the rule to study the sign of... | Let $$f(x)=\arctan\frac{x+1}{x-3}+\frac{x}4\;.$$
You know that the algebraic sign of $\arctan u$ is the same as the sign of $u$, so
$$
\arctan\frac{x+1}{x-3}\text{ is }\begin{cases}
\text{positive}&\text{when }x>3\\
\text{negative}&\text{when }-1<x<3\\
0&\text{when }x=-1\\
\text{positive}&\text{when }x<-1\;.
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
about Pythagorean quadruples Respected Mathematicians,
I would like to prepare a function, which will generate Pythagorean quadruples (a, b, c, d) = $d^2$ = $a^2$ + $b^2$ + $c^2$...-> (1). How far I am correct I don't know. For some set of integers a and b, consider $a^2$ + $b^2$ = m and d = c + ε. Now by (1), c = (m ... | You can assume that $a, b, c, d$ are relatively prime (no prime dividing them all), which in this case need not be the same as coprime (no prime dividing any pair).
This is because by homogeneity any other solution is a multiple of one that is a relatively prime. (This assumption can't be made, where the degrees are no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Use mathematical induction to prove that for all integers $n \ge 2$, $2^{3n}-1$ is not prime I had a homework due yesterday with this problem.
The TA did the problem last week in discussion but I didn't understand it.
She pulled out a $7k$ almost immediately, and I have no idea from where.
It was like, it wasn't prime ... | I’m sure that she used the following factorization
$$\begin{align*}
2^{3n}-1&=(2^3)^n-1\\
&=8^n-1^n\\
&=(8-1)(8^{n-1}\cdot 1^0+8^{n-2}\cdot 1^1+\dots+8^2\cdot 1^{n-3}+8^1\cdot 1^{n-2}+8^0\cdot 1^{n-1}\\
&=7(8^{n-1}+8^{n-2}+\dots+8^2+8+1)\\
&=7\sum_{k=0}^{n-1}8^k\;,
\end{align*}$$
which as Dylan Moreland pointed o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 0
} |
Computing $\sum_{1}^{\infty}\frac{1}{2^{2n+1}n} $ with a power series- What did I do wrong? The requested sum: $\sum_{1}^{\infty}\frac{1}{2^{2n+1}n}=
\frac{1}{2}\sum_{0}^{\infty}\frac{1}{4^nn}$
My helper- this power series: $
\sum_{0}^{\infty}\frac{x^{n}}{4^n}=\frac{1}{1-\frac{4}{x}}
$
Integration due to uniform con... | The problem with your method is that you're using primitives and not definite integrals:
$$\sum\limits_{n = 0}^\infty {\frac{1}{{n + 1}}{{\left( {\frac{x}{4}} \right)}^{n + 1}}} = - \log \left( {1 - \frac{x}{4}} \right) = - \log \left( {4 - x} \right) + \log 4 = \int\limits_0^x {\frac{1}{{4 - t}}dt} $$
Since you k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$ Given that n is a positive integer show that $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$.
I'm thinking that I should be using the property of gcd that says if a and b are integers then gcd(a,b) = gcd(a+cb,b). So I can do things like decide that $\gcd(n^3 + 1, n^2 + 2) = ... | Let $d=\gcd(n^3+1,n^2+2)$ then we have:
$d|n^3+2n\Rightarrow\ d|2n-1\Rightarrow\ d|2n^2-n\Rightarrow\ d|2n^2+4\Rightarrow\ d|n+4\Rightarrow\ d|2n+8\Rightarrow\ d|9\Rightarrow\ d=1,3,9$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/109876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 4
} |
Combining Taylor expansions How do you taylor expand the function $F(x)={x\over \ln(x+1)}$ using standard results? (I know that WA offers the answer, but I want to know how to get it myself.) I know that $\ln(x+1)=x-{x^2\over 2}+{x^3\over 3}+…$ But I don't know how to take the reciprocal. In general, given a function $... | Concerning the multiplicative inverse of $\ln(1+x)$ you may use the classical division (here $x$ is divided by $\ln(1+x)$ ) :
$\begin{array} {r|l}
x +0 x^2+0 x^3 & x-\frac{x^2}2+\frac{x^3}3+\cdots \\
-x+\frac{x^2}2-\frac{x^3}3 & 1+\frac x2-\frac {x^2}{12}\\
\frac{x^2}2-\frac{x^3}3 & \\
-\frac{x^2}2+\frac{x^3}4 & \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Using double angle formulas in integration, trouble following an example. I have just started looking at integration and I am having trouble understanding what has been done in one of the examples in the book I am working through.
It involves using the double angle formula for $\sin(2\theta)$ to provide a rearrangeme... | You know that $$\sin(2 \theta)=2\sin(\theta)\cdot\cos(\theta)$$ Now, put $\theta=\dfrac{1}{2}x$ to see that, $$\begin{align}\sin\left(2 \cdot\dfrac{1}{2}x\right)&=2\sin\left(\dfrac{1}{2}x\right)\cdot\cos\left(\dfrac{1}{2}x\right)\\\sin(x)&=2\sin\left(\dfrac{1}{2}x\right)\cdot\cos\left(\dfrac{1}{2}x\right)\\\sin\left(\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Absolute Convergence of an Infinite Product based on Weierstrass's Factor Theorem I am trying to show that $\left\{ \left( 1-\dfrac {z} {\pi }\right) e^{\left( \dfrac {z} {\pi }\right) }\right\} \left\{ \left( 1+\dfrac {z} {\pi }\right) e^{\left( -\dfrac {z} {\pi }\right) }\right\} \left\{ \left( 1-\dfrac {z} {2\pi }... | This product is quite a famous one:
We have that the Gamma function defined as:
$$\Gamma(z) = \int\limits_0^\infty e^{-t} t^{z}\frac{dt}{t}$$
is convergent for $\Re(z)>0$
It is not too difficult to prove that, fo $n\in \mathbb{N}$,
$$\lim\limits_{n \to \infty} \frac{\Gamma({z+n})}{ n^{z-1}n!}=1$$
This with the functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Solve $ \sqrt{2-2\cos x}+\sqrt{10-6\cos x}=\sqrt{10-6 \cos 2x} $ $$ \sqrt{2-2\cos x}+\sqrt{10-6\cos x}=\sqrt{10-6 \cos 2x} $$
I tried squaring and/or using $1-\cos x=2\sin^2{\frac{x}2}$, but no luck.
| We go some distance to a solution. Might as well get rid of the silly extra $2$'s, and solve
$$\sqrt{1-\cos x}+\sqrt{5-3\cos x}=\sqrt{5-3\cos 2x}=\sqrt{8-6\cos^2 x}.$$
From now on write $w$ instead of $\cos x$. So we are solving
$$\sqrt{1-w}+\sqrt{5-3w}=\sqrt{8-6w^2}.$$
Square both sides, simplify a bit. We get
$$\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/117807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Mathematical Analysis: Mean Value Theorem I'm trying to learn how the mean value theorem works by attempting to answer the following but i just dont understand it at all.
Consider the function $f(x)$ $=$ $\sqrt x$ for $x > 0$
i. Show that $f'(x)$ is a decreasing function on $(0, \infty)$
ii. Apply the Mean Value Theror... | We are given that $f(x) = \sqrt{x}$.
Hence, the first and second derivatives of $f$ are$f'(x) = \frac1{2\sqrt{x}}$ and $f''(x) = -\frac1{4x^{3/2}}$. Note that $f''(x) < 0$ for all $x \in \left(0, \infty\right)$. If the first derivative of a function is always negative, then the function is a decreasing function.
In our... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
For complex $z$, find the roots $z^2 - 3z + (3 - i) = 0$
Find the roots of:
$z^2 - 3z + (3 - i) = 0$
$(x + iy)^2 - 3(x + iy) + (3 - i) = 0$
$(x^2 - y^2 - 3x + 3) + i(2xy -3y - 1) = 0$
So, both the real and imaginary parts should = 0. This is where I got stuck since there are two unknowns for each equation. How do I p... | Use $(z-a)(z-b)=z^2-(a+b)+ab=0$ to get
$$
(a+b)=3 \tag{1}
$$
$$
ab=(3-i) \tag{2}
$$
From (1) you get $\Im(a)=-\Im(b)$. So $(a_r+ia_i)(b_r-ia_i)=(a_rb_r+a_i^2)+i(-a_r+b_r)a_i=(3-i)$, thus $a_rb_r+a_i^2=3$ and $(b_r-a_r)a_i=-1$. Now you can try a few values like ...
$a_i=1$ and figure out that $a_r=2$ and $b_r=1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/119626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find all integer solutions to $\displaystyle{2^k = 9^m+7^n}$ I can only find the trivial ones
$(m,n,k)$ as $(0,0,1), (1,1,4)$
Cannot find any more. Are there any more values? More imporantly how to show those are the only ones.
| Suppose $m>0$.
If $2^k = 9^m + 7^n$, then $2^k \equiv 1 \pmod 3$, thus $k \equiv 0 \pmod 2$.
Put $k = 2l$ : $7^n = 4^l - 9^m = (2^l - 3^m)(2^l + 3^m)$.
If $(2^l - 3^m) > 1$, then both factors are multiples of $7$, and thus so is $3^m$, which is impossible.
Thus $2^l - 3^m = 1$. The only solution to this is $l=2$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Show $ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$ Show $$ I = \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = \frac{\pi}{\sqrt 2}$$
| First notice that the function $\displaystyle{\frac{1}{1+\cos^2 x}}$ is an even function, and therefore
$$ \int_0^{\pi} \frac{\mathrm{d}x}{1+\cos^2 x} = 2 \int_0^{\frac{\pi}{2}} \frac{\mathrm{d}x}{1+\cos^2 x}$$
$$ \begin{align*}
2 \int_0^{\frac{\pi}{2}} \frac{1}{1+\cos^2 x} \,\mathrm{d}x &= 2 \int_0^{\frac{\pi}{2}} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/125637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Computing $ \iint_{[0,1]^2} \frac{-x\ln(xy)}{1-xy} \mathrm dx \mathrm dy $ I would like to compute $$ \iint_{[0,1]^2} \frac{-x\ln(xy)}{1-xy} \mathrm dx \mathrm dy $$
Without going into detail, here is what I found:
$$ \int_{0}^{1}(\int_{0}^{1} \frac{-x\ln(xy)}{1-xy} \mathrm dx ) \mathrm dy=\int_{0}^{1}(-\sum_{n=0}^{\in... | \begin{align*}\int_0^1x^{n+1}y^n\ln(xy)dx&=y^n\int_0^1x^{n+1}\ln xdx+y^n\ln y\int_0^1x^{n+1}dx\\
&=y^n\left[\frac{x^{n+2}}{n+2}\ln x\right]_0^1-y^n\int_0^1\frac{x^{n+2}}{n+2}\frac 1xdx
+y^n\ln y\frac 1{n+2}\\
&=-\frac{y^n}{(n+2)^2}+y^n\frac{\ln y}{n+2}\\
&=\frac{y^n}{n+2}\left(\ln y-\frac 1{n+2}\right)
\end{align*}
and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/129660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find the expected value of a dice sum If fair dodecahedron is rolled until at least $k$($k$ is fixed between 2 and 12) is gotten, and $X$ is the sum of all numbers appeared until the last time, what is $E(X)$?
| The probability that any roll is greater than or equal to $k$ is
$$
\frac{13-k}{12}
$$
so the expected number of rolls until a roll of $k$ or greater is
$$
\frac{12}{13-k}.
$$
All but the last one of these rolls is less than $k$, so the sum of those rolls has an expected value of
$$
\left( \frac{12}{13-k} -1 \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Differentiation of $y = \tan^{-1} \Bigl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\Bigr\}$ How do i differentiate the following: $$y = \tan^{-1} \biggl\{ \frac{\sqrt{1+x^{2}} - \sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+ \sqrt{1-x^{2}}}\biggr\}$$
I know that $\text{derivative}$ of $\tan^{-1}{x}$ is $\... | First the idea is to observe as to what would you substitute for $x$ in order to remove the square root. After some manipulations you find that the correct substitution is $x^{2} = \sin{2\theta}$. Once you have done this then you have $$y = \tan^{-1}\biggl\{ \frac{(\cos\theta + \sin\theta) - (\cos\theta - \sin\theta)}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/131679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Finding lowest value measures between three locations I've made an image of what the geography of my problem looks like:
Essentially, there are two "chemical plants" (A and B) located on a road $12$ miles apart. The pollution from plant A is given by the following equation (for some constant $K$):
$$\frac{K}{x^2 + 10}... | Choose a co-ordinate system with Plant A as origin.
Basic trigonometry to find the angles of the triangle ABC
You will get (in Radians)
$\angle A = 0.9582$
$\angle B = 0.4214$
$\angle C = 1.762$
Now the position of Plant C is straightforward $(5\cos A, -5\sin A)$
Assuming the origin is fixed at Plant A the amount of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/133947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?
Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?
$150$ degree
the right answer are $\frac{1}{2}$, $-\frac{\sqrt{3}}{2}$and $-\frac{1}{\sqrt{3}} $
$-315$ degree
the right answer are $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$ and $1$.
| It is very simple.
$\sin(150) = \sin(90+60) = \cos(60) = \frac{1}{2}$
$\cos(150) = \cos(90+60) = - \sin(60) = -\frac{\sqrt{3}}{2}$
$\tan(150) = \tan(90+60) = - \cot(60) = -\frac{1}{\sqrt{3}} $
similarly
$\sin(-315)=-\sin(270+45)=\cos(45) = \frac{1}{\sqrt{2}}$
$\cos(-315)=\cos(315)=\cos(270+45)=\sin(45) = \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/135698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
How to solve this Diophantine Equation. Step by Step What are all the solutions to the diophantine equation:
6x-6y-xy=0.
| The 18 integer solutions are :
x = -42, y = 7
x = -24, y = 8
x = -18, y = 9
x = -15, y = 10
x = -12, y = 12
x = -10, y = 15
x = -9, y = 18
x = -8, y = 24
x = -7, y = 42
x = -5, y = -30
x = -4, y = -12
x = -3, y = -6
x = -2, y = -3
x = 0, y = 0
x = 3, y = 2
x = 6, y = 3
x = 12, y = 4
x = 30, y = 5
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/136079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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What is this series called? $\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} \dots \pm \frac{x^n}{n!}$ I remember learning about this series in Precalculus the other day but I neglected to get the name of it. It looks something like this:
$
\begin{align*}
\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x... | Are you sure you didn't get the summands flipped?
The following series is called the Taylor series expansion of $\sin{x}$:
\begin{align*}
\sin{x} &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\
&= \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} \\[8pt]
\end{align*}
It's derived here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/138470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Interesting Determinant Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$.
Define \begin{equation*}
A=%
\begin{bmatrix}
0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & x_{n}-x_{1} \\
x_{2}-x_{1} & 0 & \cdots & x_{n-1}-x_{2} & x_{n}-x_{2} \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
x_{n-... | Clearly the determinant is $0$ if $x_i = x_{i+1}$ (because two adjacent rows are identical) or $x_1 = x_n$ (last row is $-$ first row). So the determinant must be a polynomial divisible by $(x_1 - x_2)(x_2 - x_3) \ldots (x_{n-1} - x_n)(x_n - x_1)$. But the determinant has degree $n$, so it is a constant times this pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/144818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Probability question (CDF , PDF etc)
Possible Duplicate:
Please correct my answer (Probability)
I have to calculate the density function of the random variable
$Y= 1-X^2$, given that: $f(x) = \frac{1}{9}(x+1)^2$, where $-1 < x < 2$.
So I finally understood that the domain of Y is $-3 < Y < 1$.
But it seems that i c... | It is easiest to start with CDF:
$$
F_Y(y) = \mathbb{P}(Y \leqslant y) = \mathbb{P}(1-X^2 \leqslant y) = \mathbb{P}(X^2 \geqslant 1 - y) = \mathbb{P}(X \geqslant \sqrt{1 - y}) + \mathbb{P}(X \leqslant -\sqrt{1 - y})
$$
Since $$F_X(x) = \begin{cases} 0 & x < -1 \\ 1 &x > 2 \\ \left(\frac{x+1}{3}\right)^3 & -1 \leqsla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$ Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
| This is a nice little question. I am assuming that $p \in \mathbb{Z}^+$, though same could be said about it when $p \notin \mathbb{Z}^+$. Before getting to the answer lets experiment a bit for small positive integers $p$. To start off, you could try for some values $p$.
For $p=1$, we get $$\lim_{n \rightarrow \infty} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 7,
"answer_id": 3
} |
Something is wrong with this proof, limit $\lim\limits_{(x,y) \to (0,0)} \frac{xy^3}{x^4 + 3y^4}$ Could someone please tell me what is wrong with this proof?
Show that $\lim\limits_{(x,y) \to (0,0)} \dfrac{xy^3}{x^4 + 3y^4}$ does not have a limit or show that it does and find the limit.
I know it is wrong because th... | The easiest way to see that a limit doesn't exist is to write
$$
\frac{xy^3}{x^4+y^4}=\frac{x/y}{(x/y)^4+1}\tag{1}
$$
Using $(1)$, it is easy to see that, over any circle around the origin, $\frac{xy^3}{x^4+y^4}$ can take on any value that $\frac{t}{t^4+1}$ can; that is, any value in $\left[{-}\sqrt[4]{\frac{27}{256}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
$\cos(x)$ and $\arccos(x)$ couple limit
Find the value of the following limit:
$$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac
{1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot
\arccos{1}}{n}$$
| A more convenient way to state the sequence is:
$$
\frac{\sum_{k=1}^n\cos\frac{1}{k}\arccos\frac{1}{n-k+1}}{n}
$$
Note that for $0\leq x\leq 1$ we have $1-x\leq\cos x\leq 1$ and $\frac{\pi}{2}-\frac{\pi}{2}x\leq\arccos x\leq \frac{\pi}{2}-x$. Therefore, we have
$$
\tfrac{\pi}{2}(1-\tfrac{1}{k})(1-\tfrac{1}{n-k+1})\leq\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/149734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Another limit related to pi number Find the value of the limit:
$$\lim_{n\to\infty} \sum_{k=0}^n \frac{{k!}^{2} {2}^{k}}{(2k+1)!}$$
I'm trying to find out if this limit can be computed only by using high school
knowledge for solving limits. Thanks.
| Mimicking robjohn's solution to the series, and after proving convergence, we may proceed as follows:
$$\sum\limits_{k = 0}^\infty {\frac{{k!^2{2^k}}}{{\left( {2k + 1} \right)!}}} = \sum\limits_{k = 1}^\infty {\frac{{\left( {k - 1} \right)!^2{2^{k - 1}}}}{{\left( {2k - 1} \right)!}}} = \sum\limits_{k = 1}^\infty {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/150512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Every number $2^N > 4$ can be expressed by the sum of 2 primes? For example the first cases are:
$2^3= 8 = 3+5$
$2^4= 16 = 3+13$
and so on ...
| 32=2^5.It is of the form 4k.we know that primes of the form 4a+1,4b-1. Now we write 32=4a+1+4b-1.then a+b=8. By solving a+b=8, we get the pairs (a,b)=(3,5),(7,1) for which 4a+1,4b-1 are primes. In this way we can find 2 primes for every power 2. For ex. 22 is of the form 4k+2. Now we write 22= 4a+1+4b+1where 4a+1 and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/151965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculating the surface area of sphere above a plane How do I calculate the surface area of the unit sphere above the plane $z=\frac12$?
EDIT: I have been attempting things and I am thinking about parameterizing this... While I know that surface area is given by the double integral of the cross products of partial deri... | Surface area is given by
$$
\iint_R \left| \vec r_u \times \vec r_v \right| \ dA
$$
where $\vec r(u,v)$ is the parametrization of the surface. We can rewrite this as (derivation shown here: http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx):
$$
\iint_D \sqrt{ \left(\frac{\partial z}{\partial x}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Finding $\int \frac {dx}{\sqrt {x^2 + 16}}$ I can not get the correct answer.
$$\int \frac {dx}{\sqrt {x^2 + 16}}$$
$x = 4 \tan \theta$, $dx = 4\sec^2 \theta$
$$\int \frac {dx}{\sqrt {16 \sec^2 \theta}}$$
$$\int \frac {4 \sec^ 2 \theta}{\sqrt {16 \sec^2 \theta}}$$
$$\int \frac {4 \sec^ 2 \theta}{4 \sec \theta}$$
$$\in... | What you have done is correct! Note that whenever you have inverse trigonometric expressions you can express your answer in more than one way! Your answer can be expressed in a different way (without the trigonometric and inverse trigonometric functions) as shown below.
We will prove that $$\sec \left( \arctan \left( \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Sum with binomial coefficients: $\sum_{k=1}^m \frac{1}{k}{m \choose k} $ I got this sum, in some work related to another question:
$$S_m=\sum_{k=1}^m \frac{1}{k}{m \choose k} $$
Are there any known results about this (bounds, asymptotics)?
| You know that $\displaystyle (x + 1)^m = \sum_{k=0}^m {m \choose k} x^k$. So
$$\int_0^1 \frac{(x + 1)^m - 1}{x} \, dx = \sum_{k=1}^m {m \choose k} \frac{1}{k}.$$
Letting $y = x + 1$ this is just
$$\int_1^2 \frac{y^m - 1}{y - 1} \, dy = \sum_{k=1}^m \frac{2^k - 1}{k}.$$
The contribution of the $-1$ terms is $-H_m \sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 7,
"answer_id": 3
} |
Indefinite integral of secant cubed $\int \sec^3 x\>dx$ I need to calculate the following indefinite integral:
$$I=\int \frac{1}{\cos^3(x)}dx$$
I know what the result is (from Mathematica):
$$I=\tanh^{-1}(\tan(x/2))+(1/2)\sec(x)\tan(x)$$
but I don't know how to integrate it myself. I have been trying some substitutions... | $\int \sqrt (1 + x^2) dx$
let $x = \tan \theta $
then $dx = \sec ^2\theta d\theta $
we have the integral is then:
$\int \sec ^3\theta d\theta $
recall:
$\tan ^2\theta + 1 = \sec ^2\theta $
and write as:
$\int \sec \theta (\sec ^2\theta )d\theta $
continue with integration by parts,
by letting:
$u = \sec \theta ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 8,
"answer_id": 3
} |
how many 5-digit numbers satisfy the following conditions How many five-digit numbers divisible by 11 have the sum of their digits equal to 30?
I am able to get the 5-digit numbers divisible by 11
and
I am also able to get the five-digit numbers whose sum of their digits equal to 30.
But i am not able to get how i can ... | A number is divisible by $11$ if the sum of the digits in the tens and thousands place minus the sum of the digits in the ones, hundreds, and ten thousands place is divisible by 11.
So take a number:
$a b c d e$
$(b+d)-(a+c+e)$ is divisible by $11$ (it may be $0$).
$a+b+c+d+e=30$.
$a,b,c,d,e$ are all integers that may ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/154949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\int \frac{dx}{x^2 - 2x} dx$ $$\int \frac{dx}{x^2 - 2x}$$
I know that I have to complete the square so the problem becomes.
$$\int \frac{dx}{(x - 1)^2 -1}dx$$
Then I set up my A B and C stuff
$$\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{-1}$$
With that I find $A = -1, B = -1$ and $C = 0$ which I know is w... | $$\begin{align} & {} \quad \int \frac{dx}{x^2 - 2x}\\ &=\int \frac{dx}{(x - 1)^2 -1}dx\\ &=\int \frac{dx}{(x - 1-1) (x-1+1)} \\ &=\int \frac{dx}{x(x-2)}\end{align}$$
The rest is easy (partial fractions):
$$\frac 12 \left[\int \frac {dx}{x-2} - \int \frac {dx}x\right]=\frac 12 \ln|x-2| -\frac 12 \ln |x| +C$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/156566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Limit exercise from Rudin: $\lim\limits_{n \to \infty} \sqrt{n^2+n} -n$ This is Chapter 3, Exercise 2 of Rudin's Principles.
Calculate $\lim\limits_{n \to \infty} \sqrt{n^2+n} -n$.
Hints will be appreciated.
| When you have a sequence that involves difference between two roots, it's often a good idea to try using the identity $(a-b)(a+b) = a^2-b^2$ to git rid of the root difference:
$$
\sqrt{a} - \sqrt{b} = \left(\sqrt{a} - \sqrt{b}\right)\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} = \frac{\left(\sqrt{a} - \sqrt{b}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/156955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Area of a revolution of $x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}$ I think my biggest problem here is I can not find a good way to find the square root in this problem
$$x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2} \ \ \ \ 1 \le x \le 2$$
$$\int_1^2 2 \pi \cdot {\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}} \sqrt{1 + (1... | My suggestion is: take it easy and take it step by step. Don't throw everything in one expression, because it is easy to make a slip:
$$x=\frac{1}{3}\left(y^2+2\right)^\frac{3}{2}$$
$$x'=\frac{1}{2}\left(y^2+2\right)^\frac{1}{2}\cdot\underbrace{ 2y}_{\text{chain rule}}=y\left(y^2+2\right)^\frac{1}{2}$$
$$x'^2=y^2\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $\lim\limits_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$ One of the previous posts made me think of the following question: Is it possible to evaluate this limit without L'Hopital and Taylor?
$$\lim_{x\rightarrow0} \frac{\tan(x)-x}{x^3}$$
| Here's a way which avoids derivatives and integrals.
Assume that we know that $\frac{\sin x}{x} \to 1$ as $x \to 0$.
Then we also know that
$\frac{1-\cos x}{x^2} = \frac12 \left( \frac{\sin(x/2)}{x/2} \right)^2 \to \frac12$.
Now,
$$
\frac{\tan x - x}{x^3} = \frac{1}{\cos x} \left( \frac{\sin x - x}{x^3} + \frac{1-\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
} |
Combining a radical and simplifying? How would I combine and simplify the following radical:
$$\sqrt {\frac{A^2}{2}} - \sqrt \frac{A^2}{8}$$
| Recall the following facts.
$$\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} \text{ whenever }a,b>0$$
$$\sqrt{x^2} = \lvert x \rvert \text{, where $x \in \mathbb{R}$}$$
$$\sqrt{ab} = \sqrt{a} \sqrt{b} \text{ whenever }a,b>0$$
Using the above, we then get that
\begin{align}
\sqrt{\dfrac{A^2}{2}} - \sqrt{\dfrac{A^2}{8}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prime as sum of three numbers whose product is a cube Good evening!
I am very new to this site. I would like to put the following material from Prof. Gandhi's note book and my observations. Of course it is little long with more questions. But, with good belief on this site, I am sending for good solutions/answers.
If ... | Here is an attempt that doesn't quite work. In this post, $\left( \frac{a}{p} \right)$ is the quadratic residue symbol.
If $\left( \frac{-3}{p} \right) = 1$, then $p$ is of the form $x^2+xy+y^2$.
If $\left( \frac{85}{p} \right) = 1$, then $p$ is either of the form $9 x^2 + 25 xy + 15 y^2$ or $3 x^2 + 25 xy + 45 y^2$.
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/158595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 5,
"answer_id": 1
} |
Proof about $z\cot z=1-2\sum_{k\ge1}z^2/(k^2\pi^2-z^2)$ In Concrete Mathematics, it is said that
$$z\cot z=1-2\sum_{k\ge1}\frac{z^2}{k^2\pi^2-z^2}\tag1$$
and proved in EXERCISE 6.73
$$z\cot z=\frac z{2^n}\cot\frac z{2^n}-\frac z{2^n}\tan\frac z{2^n}+\sum_{k=1}^{2^{n-1}-1}\frac z{2^n}\left(\cot\frac{z+k\pi}{2^n}+\cot\fr... | NOTE: This is incomplete. A tighter bound should be produced. Anyone able to do so is free to edit and add it.
For $x$ near the origin, $\cot x \sim \dfrac{1}{x}$. Since $\dfrac{1}{2^n}\to 0 $ we can use this. More precisely,
$$\frac{1}{x}-1<\cot x <\frac{1}{x} $$
Namely, we can dissect
$$z\cot z=\frac z{2^n}\cot\frac ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Compute the limit of $\frac1{\sqrt{n}}\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)^{1/n^2}$ Compute the following limit:
$$\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $$
I'm interested in almost any approaching way for this limit. Thanks.
| Another way: by Stolz-Cesaro theorem, we have
$$\begin{align}
L&=\lim_{n\to\infty} \frac{1}{n^2} \left(\sum_{k=1}^{n} k\log (k) - \frac{n^2\log(n) }{2}\right)\\
&=\lim_{n\to\infty} \frac{a_n}{b_n}=\lim_{n\to\infty} \frac{a_n-a_{n-1}}{b_n-b_{n-1}}\\
&=\lim_{n\to\infty} \frac{1}{n^2-(n-1)^2} \left(n\log (n) - \frac{n^2\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 1
} |
Help Verifying Trigonometric Identity I could really use help, hint or otherwise, in proving a trigonometric identity:
We are only allowed to work on one side of the equation.
$$\dfrac{2\sin^2(x)-5\sin(x)+2}{\sin(x)-2} = 2\sin(x)-1$$
| HINT: Factorize the numerator and cancel terms arguing why the terms you are canceling are not zero.
Move your mouse over the gray area for the answer.
$$\dfrac{2 \sin^2(x) - 5 \sin(x) + 2}{\sin(x) - 2} = \dfrac{2 \sin^2(x) - 4 \sin(x) - \sin(x) + 2}{\sin(x) - 2}\\ = \dfrac{2 \sin(x) (\sin(x) - 2) - ( \sin(x) - 2)}{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Finding solutions to $(4x^2+1)(4y^2+1) = (4z^2+1)$ Consider the following equation with integral, nonzero $x,y,z$
$$(4x^2+1)(4y^2+1) = (4z^2+1)$$
What are some general strategies to find solutions to this Diophantine?
If it helps, this can also be rewritten as $z^2 = x^2(4y^2+1) + y^2$
I've already looked at On the equ... | Here is one general approach. Since the product of the sum of two squares is itself the sum of two squares, then,
$$\tag{1}(4x^2+1)(4y^2+1) = 4z^2+1$$
is equivalent to,
$$\tag{2}(2x+2y)^2+(4xy-1)^2 = 4z^2+1$$
The complete solution to the form,
$$\tag{3}x_1^2+x_2^2 = y_1^2+y_2^2$$
is given by the identity,
$$\tag{4}(ac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/162862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Single-digit even natural number solutions to the equation $a+b+c+d = 24$ such that $a+b > c+d$
Possible Duplicate:
Two algebra questions
How to approach the below question:
How many single-digit even natural number solutions are there for the equation $a+b+c+d = 24$ such that $a+b > c+d$?
| Here is one approach you can use:
The even single-digit (presumably in base 10) natural numbers are precisely 2, 4, 6, 8.
Thus, if $y$ and $z$ are such numbers, then $4\leq y+z\leq 16$, and of course $y+z$ is even.
Break the problem up as follows: find the number of solutions $w$, $x$ to $w+x=24$ such that $w$ and $x$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Singular points of ODE My friend and I have conflicting answers and since his phone is off, I can't get his full solution and I don't understand his argument.
Consider this ODE
$$(x+1)y''+\frac{1}{x}y' + (x+3)y= 0$$
Basically what I did was divide out that $(x+1)$ on $y''$ and got
$$y''+\frac{1}{x(x+1)}y' + \frac{(x+3... | $(x+1)y''+\dfrac{1}{x}y'+(x+3)y=0$
$y''+\dfrac{1}{x(x+1)}y'+\dfrac{x+3}{x+1}y=0$
It should be noted that the positions of finite singular points are always appear at the positions that makes one of the coefficients diverge.
$\therefore$ the positions of finite singular points in this question are $x=0$ and $x=-1$ only,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Determine whether $\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$ is convergent or divergent. If convergent, find the sum. $$\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$$
It's geometric, since the common ratio $r$ appears to be $\frac{-3}{4}$, but this is where I get stuck. I think I need to do this: let $f(x) = ... | $By the given series have:
$a_{n}=\frac{(-3)^{n-1}}{4^{n}}$, $a_{n+1}=\frac{(-3)^{n}}{4^{n+1}}$
By the criterion of Dalamber have:
$A=\lim\frac{a_{n+1}}{a_{n}}=\frac{3}{4}<1$
Under this criterion we have that A<1 conclude that given series is convergent.
Since the given series is convergent exist sum of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/166097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Complex series: $\frac{z}{(z-1)(z-3)} = -3 \sum\limits_{n=0}^\infty \frac {(z-1)^n}{2^{n+2}} - \frac{1}{2(z-1)}$ for $0 < |z-1| < 2$
Show that when $0 < |z-1| < 2$,
$$\frac{z}{(z-1)(z-3)} = -3 \sum_{n=0}^\infty \frac {(z-1)^n}{2^{n+2}} - \frac{1}{2(z-1)}$$
I thought to attack this using a partial fraction decompos... | $$\dfrac{z}{(z-1)(z-3)} = \dfrac1{(z-1)} + \dfrac3{(z-1)(z-3)}$$
$$\dfrac1{z-3} = \dfrac1{(z-1) -2} = -\dfrac12 \left( \dfrac1{1-(z-1)/2}\right) = -\dfrac12 \left( \sum_{k=0}^{\infty} \left( \dfrac{z-1}2\right)^k\right)$$
Hence, $$\dfrac3{(z-1)(z-3)} = - \dfrac32 \left( \dfrac1{z-1} + \sum_{k=0}^{\infty} \left( \dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/166823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$. Let $a,b,c>0$ how to prove that :
$$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$$
I find that
$$\ \frac{ab}{a^{2}+3b^{2}}=\frac{1}{\frac{a^{2}+3b^{2}}{ab}}=\frac{1}{\frac{a}{b}+\frac{3b}{a}} $$
By AM-GM
$... | I have answered this question in a slightly different way.
Let us assume the following : $ \frac{a}{b}=x$ and $ \frac{b}{c}=y.$ This converts the above equation to a equation
with two variables. $$ f(x,y)=\frac{x}{3+x^2} + \frac{y}{3+y^2}+\frac{xy}{1+3(xy)^2}$$ Now to get a maxima or minima point of $f(x,y)$ we parti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 3
} |
Polynomial-related manipulation My question is:
Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$
Any help to solve this question would be greatly appreciated.
| Since $x^{11}+x^{10}+\ldots + x+1 = \frac{x^{12}-1}{x-1}$ we may first factorize $x^{12}-1$ and then divide by the factor $x-1$:
\begin{align*}
x^{12}-1 &= (x^6-1)(x^6+1)\\
&= (x^3-1)(x^3+1)(x^6+1)\\
&=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^2+1)(x^4-x^2+1),
\end{align*}
hence
$$x^{11}+x^{10}+\ldots +x+1 = (x^2+x+1)(x+1)(x^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$ I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating $\lim\limits_{n\to\infty} \prod\limits_{... | The last step of Andrew getting
\begin{align}\lim_{n\to \infty}\prod _{k=2}^n \left(1-\frac{1}{k^3}\right)= \frac{\cosh \frac{\sqrt{3} \pi }{2} \Gamma \left(n-\frac{i \sqrt{3}}{2}+\frac{3}{2}\right) \Gamma \left(n+\frac{i \sqrt{3}}{2}+\frac{3}{2}\right)}{3 \pi n^3 \Gamma^2 (n)}\end{align}
was a bit ambigous.
using an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 2,
"answer_id": 0
} |
$(x^n-x^m)a=(ax^m-4)y^2$ in positive integers How do I find all positive integers $(a,x,y,n,m)$ that satisfy $ a(x^{n}-x^{m}) = (ax^{m}-4) y^{2} $ and $ m\equiv n\pmod{2} $, with $ax$ odd?
| Brute force finds three solutions so far:
a=3, x=3, y=12, n=5, m=1
a=1, x=3, y=12, n=6, m=2
a=1, x=9, y=12, n=3, m=1
which are just different ways of saying that $729 - 9 = (9-4)\times12^2.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/169863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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