Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
If $a,b>0$ and $a+b=2$ , prove that $a^{2b}+b^{2a}+(\frac{a-b}2)^2\leqslant2$ If $a,b>0$ and $a+b=2$ , prove that
$$
a^{2b}+b^{2a}+(\frac{a-b}2)^2\leqslant2
$$
The equality occurs if and only if $(a,b)\sim(1,1)$ or $(a,b)\sim(2,0)$ or its cyclic permutations.
My attempt:
By symmetry and the constraint, we may let $x=a-... | Following your approach, WLOG, let $a=1-x, b=1+x$ for some non-negative real $x \in [0, 1)$. Then we have to show: $f = (1-x)^{2+2x}+(1+x)^{2-2x}+x^2 \leqslant 2$
Using the case of Bernoulli's inequality where exponent is in $(0,1)$, we have $(1-x)^{1+x} \leqslant (1-x)(1-x^2)$.
Again by Bernoulli's inequality, $(1+ x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4624313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
What is the probability f(p) of A winning the match?
Here's my approach:
Let A denote the event "A wins the game" and B denote the event "B wins the game"
I am supposed to calculate the probability f(p)
There are two possible cases:-
i) A wins within 4 games
Possible cases: AAA,ABAA,BAAA,AABA
ii) A doesn't win within ... | Let $q = 1 - p.$
Suppose that the match is tied 2-2 after 4 games.
Let $R$ denote the probability of A's winning the match, in that event.
Then
$$R = p^2 + 2pqR \implies R(1 - 2pq) = p^2 \implies R = \frac{p^2}{1 - 2pq}.$$
After $4$ games either A has won all $4$, A has won $3$ out of $4$ (where it is harmless to play... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4627694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
An urn has 2 white, 3 red and 5 black balls. Problem An urn has 2 white, 3 red and 5 black balls. 3 balls are randomly drawn, one at a time and without replacement.
Calculate the probability of extracting the sequence of colors (white, black, red) knowing that you have extracted a black ball.
solution
Let E be the even... | Start by computing the probability of drawing a black ball. The probability that no black ball is drawn is $\frac{1}{2} \frac{4}{9} \frac{3}{8} = \frac{1}{12}$. Therefore, the probability that a black ball is drawn is $1 - \frac{1}{12} = \frac{11}{12}$.
Now, compute the probability of observing the sequence (white, bla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4630946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to prove that $\sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$ is discontinuous at $0$?
How to prove that $\sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$ is discontinuous at $0$?
As we know, $\sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$ converges uniformly on $[\delta,\pi-\delta]$ for $\delta>0$. and not uniformly on $(0... | To establish the discontinuity of $S(x)$ at $x = 0$, it suffices to show that $S(\pi/N) \to \infty$ as $N \to \infty$. Indeed, by grouping the terms into $N$ consecutive terms, the defining sum for $S(\pi/N)$ is recast as
\begin{align*}
S(\pi/N)
&= \sum_{r=1}^{N} \sin\left(\frac{r\pi}{N}\right) \sum_{q=0}^{\infty} \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $\arcsin$ of dual number: $\arcsin\left( x+y\cdot\varepsilon\right)$ where $x,y\in\mathbb{R}$ and $\varepsilon\ne0=\varepsilon^{2}$? How to calculate $\arcsin$ of dual number:
$\arcsin\left( x + y \cdot \varepsilon \right) = \dots$
where
$\begin{align*}
a, ~b &\in \mathbb{R}\\
\varepsilon^{2} &= 0\\
\v... | "Automatic differentiation" is the way to go. Namely, for any real-analytic function $f\colon \Bbb R \to \Bbb R$, its natural extension to $\Bbb R\oplus \Bbb R \varepsilon$ in fact satisfies $$f(x+y\varepsilon) = f(x) + yf'(x)\varepsilon,$$for all $x,y\in \Bbb R$. To see this, write $$f(x) = f(0) + f'(0)x + \frac{f''(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4632042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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$\int_{0}^{\pi/2}\ln^3(\sin x)\,\mathrm dx$ There are closed forms for $\int_{0}^{\pi/2}\ln(\sin x)\,\mathrm dx\,$ and $\,\int_{0}^{\pi/2}\ln^2(\sin x)\,\mathrm dx\,$ but I can’t seem to find a closed form for $$\int_{0}^{\pi/2}\ln^3(\sin x)\,\mathrm dx\;.$$
How would I calculate it?
| Noting that $$
\int_0^{\frac{\pi}{2}} \ln ^3(\sin x) d x=\left.\frac{\partial^3}{\partial a^3} I(a)\right|_{a=0}
$$
where
$$
I(a)=\int_0^{\frac{\pi}{2}} \sin ^a x d x=\frac{1}{2}B\left (\frac{a+1}{2},\frac{1}{2} \right)$$
\begin{aligned}
&\quad \frac{\partial^3}{\partial x^3}(\mathrm{~B}(x, y)) \\&=\left[\left(\psi^{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4633165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integral inequality $I_2(z) = \mathcal{O}\left(\frac{1}{(1+z)^2}\right)$ I want to prove the inequality of an integral like this:
$$I_2(z) = \int_1^{+\infty} \frac{\rho(t)}{(z+t)^2} d t = \mathcal{O}\left(\frac{1}{(1+z)^2}\right)$$ , when $z \to +\infty$
Below is my current attempt：
$$\begin{aligned}
I_2(z) & =\int_1^{... | For $z=x+iy$ with $x > 0$ is, as you already calculated,
$$
I_2(z) = \int_1^\infty \frac{2R(t)}{(z+t)^3} \, dt
$$
and $|2 R(t)|\le 1$. Let us first assume that $y \ne 0$. Then
$$
\begin{align}
|I_2(z)| &\le \int_1^\infty \frac{1}{(x+t)^2+y^2)^{3/2}} \, dt \\
&= \left[ \frac {x+t}{y^2 \sqrt{(x+t)^2+y^2}}\right]_{t=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4636566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find the $n$th power of a 3-by-3 circulant matrix Consider the matrix given as $$A=\begin{bmatrix}a_0 & a_2 & a_1\\ a_1 & a_0 & a_2\\ a_2 & a_1 & a_0\end{bmatrix}$$
Write down a formual for $A^n$ for $n\in\mathbb{N}$.
$$$$
My attempt: The first that comes to mind is to diagonalize it and hence find the formual for $A^n... | Circulant matrices are related to Fourier transform.
Consider the DFT matrix of length $N=3$
$$
\mathbf{W}
=
\begin{pmatrix}
1 & 1 & 1 \\
1 & \Omega & \Omega^2 \\
1 & \Omega^2 & \Omega^4
\end{pmatrix}
$$
where
$\Omega
= e^{-2\pi i/N}$.
Consider now the DFT of the signal
$$
\mathbf{y}=
\mathbf{W}
\begin{pmatrix}
a_0 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4639166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$ Let $n$ be a positive integer and $x_1,...,x_n$ positive reals such that $x_1+...+x_n=1$. Prove that $$\prod_{i=1}^n\left(\frac{1}{x_i^2}-1\right)\ge (n^2-1)^n$$
Now notice that if we apply Jensen's Inequality to $$f(x)=\ln\left(\frac{1}{x_i^2}-1\rig... | WLOG, assume that $x_1 \le x_2 \le \cdots \le x_n$.
We split into two cases:
Case 1: $x_n < \frac{1}{\sqrt 3}$
Note that $x \mapsto \ln(\frac{1}{x^2} - 1)$ is convex on $(0, \frac{1}{\sqrt 3})$.
We have
$$\sum_{i=1}^n \ln\left(\frac{1}{x_i^2} - 1\right) \ge n\ln\left(\frac{1}{(\frac{x_1 + x_2 + \cdots + x_n}{n})^2} - 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4639980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $\int \frac{2x}{(1-x^2)\sqrt{x^4-1}}dx$ How do I evaluate $\int \frac{2x}{(1-x^2)\sqrt{x^4-1}}dx$ ?
Note that this is a Q&A post and I've presented my solution below.
| Letting $x^2=\sec \theta$ transform the integral into
\begin{aligned}
I & =\int \frac{\sec \theta \tan \theta d \theta}{(1-\sec \theta) \tan \theta} \\
& =\int \frac{1}{\cos \theta-1} d \theta \\
& =\int \frac{\cos \theta+1}{-\sin ^2 \theta} d \theta \\
& =-\int \cot \theta \csc \theta d \theta-\int \csc ^2 \theta d \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4640836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
$(a+1)x^{2} +(b+1)y^{2} +(c+1)z^{2} \geq 2(xy+yz+zx)$ Prove $(a+1)x^{2} +(b+1)y^{2} +(c+1)z^{2} \geq 2(xy+yz+zx)$ for $a,b,c>0$ and $abc\geq1$.
My try: $(a+1)x^2-4xy+(b+1)y^2+(a+1)x^2-4xz+(c+1)z^2+(b+1)y^2-4yz+(c+1)z^2\geq0$
Then I had to prove that $(a+1)x^2-4xy+(b+1)y^2\geq2x^2-4xy+2y^2$, but I realized that there's ... | *
*Clearly we just need to focus on $ abc = 1$. Suppose $x, y, z$ are fixed. How can we minimize the $LHS$ subject to $ abc = 1$?
*If $c$ is fixed, then we have $(a+1)x^2 + (\frac{1}{ac} + 1) y^2 + (c+1) z^2$, and differentiating with respect to $a$ gives $ x^2 - \frac{1}{c a^2} y^2 = 0 $, or that $ \frac{x^2}{y^2} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4641220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the maximum value of $a^2(b-c)+b^2(c-b)+c^2(1-c)$, if $0\le{a}\le{b}\le{c}\le{1}$.
How do I find the maximum value of $a^2(b-c)+b^2(c-b)+c^2(1-c)$, if $0\le{a}\le{b}\le{c}\le{1}$?
I have tried plugging in values, tried manipulating and inequality by means but that didn't worked. How do I approach this question?
| $$\underbrace{(c-b)}_{\ge0}\underbrace{(b^2-a^2)}_{\ge0}+c^2\underbrace{(1-c)}_{\ge0}\\
$$
$b^2-a^2$ is the only term dependent on $a$ and thus it attains a maximum at $a=0.$ The problem then reduces to
$$b^2(c-b)+c^2(1-c)$$ for $0\le b\le c\le1.$ Sub $b=rc.$
$$r^2c^3(1-r)+c^2(1-c)\\
c^2(r^2(1-r)c+1-c)$$
$r^2(1-r)\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4641649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Fraction of prime number between consecutive powers of two A $n$-bit integer is an integer $x$ such that $2^{n-1} \le x < 2^n$. In [1] it is claimed without proof that a corollary of the prime number theorem is that:
For any $n > 1$, the fraction of $n$-bit integers that are prime is at least $\frac{1}{3n}$.
Can some... | According to
the classic article
"Explicit Bounds for Some Functions of Prime Numbers",
Barkley Rosser
American Journal of Mathematics
Vol. 63, No. 1 (Jan., 1941), pp. 211-232 (22 pages)
for $x \ge 55$,
$\dfrac{x}{\log x+2}
\lt \pi(x)
\lt \dfrac{ x}{\log x-4}
$.
Therefore,
putting $x=2^n$,
$\dfrac{ 2^n}{n\log 2+2}
\lt ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4643160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Derivation of $\sin(15^\circ)$ geometrically This is my attempt:-
Let us consider a right $\triangle ABC$ such that angle $A$ is $15^\circ$ and $C$ is $75^\circ$.
On the line $AB$, let us assume a point $D$ such that $\frac{BC}{BD} =\frac{ 1}{\sqrt{3}}$ (Without Loss Of Generality). So $\angle BDC$ becomes $30^\circ$ ... | There is another method that starts with a fun and challenging geometry problem. Not sure if you've seen it before, but it is fairly famous (I don't know if it has a specific name though):
Now, once you've solved that purely with elementary geometry, no trig, you can then get some nice trig results from that. $M$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4643485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to evaluate $\int x \sqrt{x^2 - x}\ dx$
Problem is to integrate $ x \sqrt{x^2 - x}$.
My attempt:
I made it ready for a substitution $u = x^2 - x$
$$\begin{aligned} \int x \sqrt{x^2 -x}\ dx &= \int (2x-1)\sqrt{x^2 - x} \ dx - \int(x - 1)\sqrt{x^2 - x}\ dx\\& = \int \sqrt u\ du - \int(x - 1)\sqrt{x^2 - x}\ dx\\& ... | Substitute
$$t=\sqrt{x^2-x}-x \implies x = -\frac{t^2}{2t+1} \implies dx = -\frac{2t^2+2t}{(2t+1)^2} \, dt$$
and the resulting integrand is primed for partial fraction expansion.
$$\int x \sqrt{x^2-x} \, dx = 2 \int \frac{t^4(t+1)^2}{(2t+1)^4} \, dt$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4646982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Proving Generating Function holds a specific recurrence. Consider the generating function $$\dfrac{1}{1 − 2x − x^2} = \sum_{n=0}^{\infty}a_nx^n$$
Prove that for each integer $n \ge 0$,
$$a_n^2+a_{n+1}^2 = a_{2n+2}$$
Hint: Find a $2 \times 2$ matrix $A$ such that
$$A^{n+2} =\begin{bmatrix}a_n &a_{n+1}\\
a_{n+1} &a_{n+2}... | Too long for a comment
$$\frac{1}{1 − 2x − x^2}=\frac{1}{(1+\sqrt2+x)(\sqrt2-1-x)}=\frac{1}{2\sqrt2}\left(\frac{1}{1+\sqrt2+x}+\frac{1}{\sqrt2-1-x}\right)$$
$$=\frac{1}{2\sqrt2}\sum_{n=0}^\infty\left(\frac{x^n}{(\sqrt2-1)^{n+1}}+(-1)^n\frac{x^n}{(\sqrt2+1)^{n+1}}\right)$$
Then, given that $\frac{1}{(\sqrt2+1)^{n+1}}=\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4648320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the exact value of $\int_{0}^{2}x[\frac{1}{x}]dx$. Find the exact value of $\int_{0}^{2}x[\frac{1}{x}]dx$.
Let $[x]$ denote $\lceil{x-\frac{1}{2}}\rceil$.
Using Desmos, I got $2.46736022133$ and WolframAlpha does not give me a solution. My intuition tells me that it might be possible to find an exact value using T... | An alternate solution:
$$
\begin{align}
\int_{0}^{2}x\operatorname{round}\left(\frac{1}{x}\right)dx &= \int_0^2 x\Biggl\lfloor{\frac{1}{x}+\frac{1}{2}\Biggr\rfloor}dx \tag{1}\\
&= 8\int_{1}^{\infty}\frac{\lfloor{x\rfloor}}{\left(2x-1\right)^{3}}dx \tag{2}\\
&= 8\sum_{n=1}^{\infty}n\int_{n}^{n+1}\frac{dx}{\left(2x-1\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4650684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
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Is there an integral that proves $\pi > 333/106$? The following integral,
$$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$
is clearly positive, which proves that $\pi < 22/7$.
Is there a similar integral which proves $\pi > 333/106$?
| Another solution is given by the integral
$$
0 < \int_0^1 \frac{x^4(1-x)^8}{4(1+x^2)}dx = \pi -\frac{2419}{770} = \pi - \frac{333}{106}-\frac{1}{20405}
$$
This proves the stricter condition
$$
\pi > \frac{333}{106}+\frac{1}{20405}
$$
which implies
$$
\pi > \frac{333}{106}
$$
Similarly, for the fourth convergent (formul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "129",
"answer_count": 6,
"answer_id": 0
} |
Proof that $1+2+3+4+\cdots+n = \frac{n\times(n+1)}2$ Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ?
| Here are two ways to calculate this sum. First is by symmetry of another sum:
$\begin{aligned} \displaystyle & \sum_{0 \le k \le n}k^2 = \sum_{0 \le k \le n}(n-k)^2 = n^2\sum_{0 \le k \le n}-2n\sum_{0 \le k \le n}k+\sum_{0 \le k \le n}k^2 \\& \implies 2n\sum_{0 \le k \le n}k = n^2(n+1) \implies \sum_{0 \le k \le n}k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "136",
"answer_count": 36,
"answer_id": 16
} |
If $AB = I$ then $BA = I$
If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
*
*Elementary row operations.
*Linear dependence.
*Row reduced forms and their relations with the original matrix.
If the... | An alternative.
Let $\textbf{A}$ be a $n \times n$ matrix, and let $\lambda_k$ be the eigenvalues, then we can write
$$
\prod_{k=1}^n \Big( \textbf{A} - \lambda_k \textbf{I} \Big) = 0
$$
So we obtain
$$
\textbf{A}^n = a_0 \textbf{I} + \sum_{k=1}^{n-1} a_k \textbf{A}^k
$$
Let us write
$$
\textbf{B} = b_0 \textbf{I} + \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "379",
"answer_count": 34,
"answer_id": 2
} |
Explain why calculating this series could cause paradox? $$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - (1 + \frac{1}{2} + \frac{1}{3}... | In this chain of 4 equations, #1, #3 and #4 are correct, and no.2 is the mistake. The equations are assertions about (limits of) some finite sums. Let $H_n = \Sigma_{j=1}^n 1/j$ and $A_n = \Sigma_{i=1}^n (-1)^{i-1} 1/i$.
The correct formula is $A_n = H_n - H_{[n/2]}$ (which is approximately $\log(n) - \log(n/2) \sim \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Factoring a Cubic Polynomial I've been trying to understand how
${x^3-12x+9}$
factors to
$(x-3) (x^2+3 x-3)$
What factoring rule does this follow? The net result seems to be similar to what is attained through the sum/difference of cubes factoring pattern, but the signs are different.
Additionally, what type of pro... | In order to factor any cubic, you must find at least one root. You acknolwedged that $3$ is a root, thus $x=3$ and $x-3=0$. And since $x-3$ is a factor of $x^3-12x+9$, split the polynomial in accordance with $x-3$ and factor as follows:
\begin{align}
x^3-12x+9 &= x^3-3x^2+3x^2-9x-3x+9\\
&=x^2(x-3)+3x(x-3)-3(x-3)\\
&=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 7,
"answer_id": 3
} |
How to prove an identity in radicals? (4 / (3 - sqrt(5))) ^ 2 - ((6 - 5 * sqrt(6)) / (5 - sqrt(6))) ^ 2 = 2 * sqrt(61 + 24*sqrt(5))
$$\left(\frac{4}{3-\sqrt{5}}\right)^2 - \left(\frac{6 - 5 \sqrt{6}}{5 - \sqrt{6}}\right)^2 = 2\sqrt{61+24\sqrt{5}}$$
How to prove it is right equality?
I come up with $\dfrac{16}{14-6\sqrt... | Specialize $\rm\ \ \ b=3,\ \ c = 5,\ \ d = 6 \ \Rightarrow\ a = 4\ \ $ in this simple derivation:
$\rm\quad\quad\quad\quad\displaystyle \bigg(\frac{b^2-c}{b-\sqrt{c}}\bigg)^2 - \bigg(\frac{d -5\sqrt{d}}{5-\sqrt d}\bigg)^2$
$\rm\quad\quad =\quad \:(\: b \ \: + \: \sqrt{c}\ )^{\:2} \ \ \:-\ \ \ (\:-\:\sqrt{d}\:)^{2} $
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$. If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
My attempt at answering the question:
\begin{align*}
x &\equiv 0 \pmod{a}\\\
&\Longrightarrow x\text{ is divisible by $a$}\\\
&\Longrightarrow x = ma\text{ for some integer $m$}\\\
\ \\\
x &\equiv 0 \pmod{b}\\\
&\Lon... | You are given that $a$ divides $x$. Therefore, you can write $x = ma$ where m is an integer. You are also given that $b$ divides $x$. This implies that $b$ divides $ma$. But $b$ and $a$ are coprime. Therefore $b$ must divide $m$. So you can write $m = kb$ where $k$ is another integer. Therfore, you have $x = kab$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/8243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
The Basel problem As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it w... | Let $f(x)=\frac 12-x$ on the interval $[0, 1)$, and extend $f$ to be periodic on $ \mathbb{R} $.
By definition,
\begin{align*}
\hat f(0)=\int_0^1 f(x)dx=\int_0^1 \left(\frac 12-x\right)dx=0.
\end{align*}
And for $ \kappa\ne 0 $:
\begin{align*}
\hat f(\kappa)&=\int_{0}^{1}f(x)e^{-2\pi i\kappa x }dx=\int_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "814",
"answer_count": 48,
"answer_id": 27
} |
Probability and Integrals Suppose $f(x,y) = c$ for $0\lt y\lt x\lt 1$ and $0$ outside. What is $P(X+Y \leq 1)$? What is $P(X^2+Y^2 \leq 1)$?
So
\begin{equation*}
P(X+Y \leq 1) = \int_{0}^{1} \int_{0}^{1-x} 2 \ dy \ dx?
\end{equation*}
Likewise,
\begin{equation*}
$P(X^2+Y^2 \leq 1) = \int_{0}^{1} \int_{-\sqrt{1-x^2}}... | Since the joint density is constant on the support region, you can get your answers by considering areas. The answer in each case is the area that corresponds to the event in question divided by the total area of the support region.
Thus, to calculate $P(X + Y \leq 1)$, you want the area of the lower triangular regi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/8526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Expected number of neighbors Given a row of 16 houses where 10 are red and 6 are blue, what is the expected number of neigbors of a different color?
| The chances of a particular neighbour pair being the same colour is
$$ \frac{{14 \choose 8} + {14 \choose 10}}{{16 \choose 6}} = \frac{4004}{8008} = \frac{1}{2}$$
Hence the answer is $\displaystyle 7.5$
This is happening because $\displaystyle {14 \choose 8}, {14 \choose 9}, {14 \choose 10}$ are in arithmetic progessio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Prove by induction $T(n) = 2T(\frac{n}{2}) + 2$ I'm stuck with this induction proof:
So far, given:
$\begin{align*}
T(1) & = 2 \\
T(n) & = 2T(n/2)+2 \\
& = 2(2T(n/[2^2])+2) + 2 \\
& = [2^2]T(n/[2^2]) + [2^2] + 2 \\
& = [2^2](2T(n/[2^2])+2) + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + [2^3] + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + ... | HINT $\: $ From the first few values we guess $\rm\ T(2^n)\ =\ 2^{n+2}-2\ $ and induction confirms it:
$$\rm T(2^{n+1})\ =\ 2\ T(2^n) + 2 \ =\ 2\ (2^{n+2}-2) +\ 2\ =\ 2^{n+3} - 2$$
One can extend $\rm\:T\:$ to $\:\mathbb N\:$ by defining $\rm\ T(2k+1) = 2\ T(k+1)-2 $ and now one easily proves by induction that $\rm\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Can I find the limit of $\sum_\limits{i=1}^{n^2}\frac{1}{\sqrt{n+i}}$ with the squeeze theorem? Problem: Calculate limit of $\frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \frac{1}{\sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n+n^2}}$ as $n$ approaches infinity.
Solution: Denote the above some as $X$, then we can bound it:
$$ ... | The term $\frac{1}{\sqrt{n+n^2}}$ goes to zero as $n \rightarrow \infty$, so that side doesn't work. You should think about how many terms you are adding.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/10745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
How to derive the following series expansion for $\ln \left( \frac{x+1}{x-1} \right)$? How to prove :
$$ \ln \biggl(\frac{x+1}{x-1}\biggr) = 2\biggl[\frac{1}{x} + \frac{1}{3x^3} + \frac{1}{5x^5} + \cdots \biggr]$$ where $|x| \gt 1$
I am not able to get how to proceed on this one ?
| If $|x|>1$ then $\frac{1}{|x|}<1$, so:
$$\ln\left(\frac{x+1}{x-1}\right)=\ln\left(\frac{x(1+\frac{1}{x})}{x(1-\frac{1}{x})}\right)=\ln\left(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}\right)=\ln(1+t)-\ln(1-t)$$
With $t=\displaystyle\frac{1}{x}<1$, then $|t|<1$
Expanding :
$$\ln(1+t)=t-\frac{t^{^2}}{2}+\frac{t^{^3}}{3}-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
Possible Duplicate:
Value of $\sum x^n$
Proof to the formula
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
| Observe that
\begin{eqnarray}
x^{n+1} - 1 = x^{n+1} + (x^{n} - x^{n}) + \cdots + (x - x) - 1 = (x^{n} + x^{n-1} + \cdots + x + 1)(x - 1).
\end{eqnarray}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/11703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 3
} |
Show that $\gcd(7^{79}+5,7^{78}+3) = 4$ How can I prove that $\gcd(7^{79}+5,7^{78}+3) = 4$ ? This was a question on a past exam, so the naive euclidean algorithm doesn't seem to suffice.
I'm not really sure where to start with this.
Note: This is exam prep, not homework.
| Since $\gcd(a,b)=\gcd(a-b,b)$, $$\begin{align}
\gcd(7^{79}+5,7^{78}+3)
&=\gcd(7\cdot 7^{78}+5-7^{78}-3,7^{78}+3)
\\
&=\gcd(6\cdot 7^{78}+2,7^{78}+3)
\\
&=\gcd(6\cdot 7^{78}+2-7^{78}-3,7^{78}+3)
\\
&=\gcd(5\cdot 7^{78}-1,7^{78}+3)
\\
&\vdots
\\
&=\gcd(7^{78}-13,7^{78}+3)
\\
&=\gcd(7^{78}-13,7^{78}+3-7^{78}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/13608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Algebra Problem The expression $x^2-4x+5$ is a factor of $ax^3+bx^2+25$. Express the sum a+b as an integer.
Please give an explanation of how the answer
| If $x^2-4x+5$ is a factor of $ax^3+bx^2+25$, then $ax^3+bx^2+25=(x^2-4x+5)(\text{something})$. Since $ax^3+\cdots$ is a polynomial of degree 3 and $x^2-\cdots$ is a polynomial of degree 2, the "something" must be a polynomial of degree 1:
$$ax^3+bx^2+0x+25=(x^2-4x+5)(\underline{\;\;\;\;\;\;}x+\underline{\;\;\;\;\;\;})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/16284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Derivative of a rational function I've found the following derivative in my Calculus book and I can't get my my head around the algebra involved. Can anybody help me?
Thanks.
| \begin{align*}
\frac{d}{dx}\left(-\frac{x^2+1}{(x^2-1)^2}\right) &= -\frac{d}{dx}\left(\frac{x^2+1}{(x^2-1)^2} \right)&\quad&\mbox{(1)}\\
&= -\left(\frac{(x^2-1)^2(x^2+1)' - (x^2+1)\left((x^2-1)^2\right)'}{\left((x^2-1)^2\right)^2}\right)&&\mbox{(2)}\\
&= - \frac{(x^2-1)^2(2x) - (x^2+1)\left(2(x^2-1)(x^2-1)'\right)}{(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Factoring $a^{10}+a^5+1$ I'm very interested to know how I can factorise $a^{10} +a^5 +1$ in two factors with integer coefficients. I've tried a lot but I don't have any idea how do that.
| Another hint: can you factor $a^{15}-1$? In more than one way?
Six years later, for the record, let's spell out what I meant by this. What I had intended was to observe
$$a^{15}-1 = (a^3)^5 - 1 = (a^3-1)(a^{12} + a^9 + a^6 + a^3 + 1)
= (a-1)(a^2+a+1)(a^{12} + a^9 + a^6 + a^3 + 1)$$
and also
$$a^{15}-1 = (a^5)^3 - 1 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/24529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
How can I find $\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$? My question is, how can I evaluate the following integral?
$$\int\frac1{\sqrt[4]{1+x^4}}\mathrm dx$$
Thanks.
| Integral
$$\begin{align*}
\int\frac{1}{(1+x^4)^{1/4}}dx&=\int \frac{1}{x(1+1/x^4)^{1/4}}dx\\
&=\int \frac{x^4}{x^5(1+1/x^4)^{1/4}}dx.
\end{align*}$$
Substitution: $z^4=(1+1/x^4)$
$4z^3 dz=-4\frac{1}{x^5}dx$
Therefore,
$$\begin{align*}
\int\frac{1}{(1+x^4)^{1/4}}dx&=\int \frac{x^4}{x^5(1+1/x^4)^{1/4}}dx\\
&=-\int\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/30307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Where is the mistake in this incorrect proof in Eisenstein integers? The Diophantine equation is $n^2 + n + 1 = m^3$ my attempt to solve it shows there is no solutions to this equation, but in fact there are four. I could not find my mistake so I hope someone could point it out please.
Factoring in the Eisenstein inte... | From
$$\mbox{cube}= -3 \frac{n - \omega}{1+2 \omega}\frac{n + 1 + \omega}{1 + 2 \omega}$$ doesn't the inference
"so either $n−\omega\ $ or $n+1+\omega\ $ is $1+2\omega\ $ times a cube"
also need $-3$ to be prime? This is not true in the Eisenstein integers.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/30518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How can I can solve integrals of rational functions of polynomials in $x$? I would like to know a way to solve integrals such as this one:
$$\int \frac{x}{3x - 4}dx$$
Also, I assume similar integrals where x is squared are solved in a similar manner. (If the answer is yes then don't also show me how to solve this secon... | Whenever you are trying to integrate a rational function, the first step is to do the division so that the numerator is of degree strictly smaller than the numerator (this is what Eugene Bulkin and J.M. are saying in the comments). For example, for
$$\int \frac{x}{3x-4}\,dx$$
you should do the division of $x$ by $3x-4$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/32095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Simpler way to compute a definite integral without resorting to partial fractions? I found the method of partial fractions very laborious to solve this definite integral :
$$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$
Is there a simpler way to do this ?
| By using techniques of complex analysis ($\text{Residue Theory}$) one can actually show that $$\int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx} = \frac{\pi}{b \sin(\pi{a}/b)}, \qquad 0 < a <b$$
You can obtain the value of your $\text{Integral}$ by putting $a=\frac{4}{3}$ and $b=2$.
Set $$I = \int\limits_{0}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/34351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 8,
"answer_id": 5
} |
A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$? I am currently trying to show that $\int_{-\infty}^\infty \cos(x^2) \, \mathrm dx = \sqrt{\frac{\pi}{2}}$ and the last integral I have to evaluate is
$$\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx.$$
Now of course I'm familiar with wolframalpha, how... | $\int_{-a}^a\dfrac{x^2}{x^4+1}~dx$
$=\int_{-a}^a\dfrac{1}{x^2+\dfrac{1}{x^2}}~dx$
$=\dfrac{1}{2}\int_{-a}^a\dfrac{1+\dfrac{1}{x^2}}{x^2+\dfrac{1}{x^2}}~dx+\dfrac{1}{2}\int_{-a}^a\dfrac{1-\dfrac{1}{x^2}}{x^2+\dfrac{1}{x^2}}~dx$
$=\dfrac{1}{2}\int_{-a}^a\dfrac{1+\dfrac{1}{x^2}}{\left(x-\dfrac{1}{x}\right)^2+2}~dx+\dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/37229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 9,
"answer_id": 8
} |
Where are these additional solutions coming from? Solve for $x$: $2\sin(2x)-\sqrt{2} = 0$ in interval $[0,2\pi)$
Step $1$: Add $\sqrt{2}$ and divide by $2$ to get $\sin(2x) = \dfrac{\sqrt{2}}{2}$
Step $2$: Set $2x$ equal to the angles where $\sin(x) = \dfrac{\sqrt{2}}{2}$:
$2x = \dfrac{\pi}{4}$ and $2x = \dfrac{3\pi... | Your step two is correct except for a minor omission. More properly,
$2x = \displaystyle \frac{\pi}{4} + 2k\pi$, $k$ integer, and
$2x = \displaystyle \frac{3\pi}{4} + 2k\pi$, $k$ integer
Dividing these two expressions by 2 yield
$x = \displaystyle \frac{\pi}{8} + k\pi$, $k$ integer, and
$x = \displaystyle \frac{3\pi}{8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/38583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Solve $\sqrt{3}\tan\theta=2\sin\theta$ I am trying to solve $\sqrt{3}\tan\theta=2\sin\theta$ on the interval $[-\pi,\pi]$.
$$\sqrt{3}\tan\theta=2\sin\theta \Rightarrow \sqrt{3}=\frac{2\sin\theta}{\tan\theta}$$
$$\Rightarrow \sqrt{3}=2\sin\theta \cdot \frac{\cos\theta}{\sin\theta} \Rightarrow 3 = 4\cos^2\theta$$
I get $... | $$\sqrt{3}\tan\theta=2\sin\theta$$
$$\sqrt{3}\frac{\sin\theta}{\cos\theta}=2\sin\theta$$
$$\sqrt{3}\sin\theta-2\sin\theta\cos\theta=0$$
$$\sin\theta(\sqrt3-2\cos\theta)=0$$
$$\sin\theta=0, \ \ \cos\theta=\frac{\sqrt3}{2}$$
$$\theta=k\pi , \ \ \ \theta=2k\pi\pm \frac{\pi}{6}$$
For given interval, $\theta\in[-\pi, \pi]$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
Define term using Stirling Numbers I am trying to solve the following exercise - quite unsuccessful yet.
Let a(m,n) be defined as
$$ \sum\limits_{n=0}^m a(m,n) \prod\limits_{i=1}^n (x+i-1) = x^m $$
Express a(m,n) using S(m,n) while S(m,n) are the Stirling numbers of the second kind which count the number of ways... | The shortest way to find the answer is to replace $x$ by $-x$ in one of the identities and then compare them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/40492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
The sum of $(-1)^n \frac{\ln n}{n}$ I'm stuck trying to show that $$\sum_{n=2}^{\infty} (-1)^n \frac{\ln n}{n}=\gamma \ln 2- \frac{1}{2}(\ln 2)^2$$
This is a problem in Calculus by Simmons. It's in the end of chapter review and it's associated with the section about the alternating series test. There's a hint: refer to... | Consider
$$P=\frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+...+\frac{\ln 2n}{2n}$$
$$Q=\frac{2\ln 2}{2}+\frac{2\ln 4}{4}+\frac{2\ln 6}{6}+...+\frac{2\ln 2n}{2n}=\left( 1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n} \right)\ln 2+\frac{\ln 2}{2}+\frac{\ln 3}{3}+\frac{\ln 4}{4}+...+\frac{\ln n}{n}$$
$$Q-P=\sum\limits_{j=2}^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/40998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 5,
"answer_id": 4
} |
Solving $\sqrt{x+5} = x - 1$ I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it:
$$\sqrt{x+5} = x - 1$$
So I used this logic:
$$
\begin{align}
\sqrt{x+5} &= x - 1 \\
x + 5 &= (x-1)^2 \\
x + 5 &= (x-1)(x-1) \\
x + 5 &= x^2 - 2x + 1 \\
0 ... | At no stage...
Here is what happens, you squared the equation
$$\sqrt{x+5} =x-1 \,.$$
But then, if the two sides of the equation have the same absolute value, but opposite signs they are not equal in this equation but they become equal at the next step.
This means that the equation $x+5=(x-1)^2$ possibly has more solu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 7,
"answer_id": 3
} |
Limit of eigenvectors versus eigenvectors of limit Consider the matrix
$$
A=\begin{pmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & g
\end{pmatrix},
$$
where $g$ is a real parameter. If I set $g=0$ and calculate the normalized eigenvectors of $A|_{g=0}$ with Mathematica, I find that they are
$$
v_1 = \frac{1}{\sqrt... | Both results are correct. $u_1$ and $u_2$ correspond to the same eigenvalue $-1$, and $\left( \matrix{-1\cr 0\cr 1\cr} \right) = \frac{1}{2} \left(\matrix{-1 \cr 1 \cr 0\cr} \right) + \frac{1}{2} \left(\matrix{-1 \cr -1 \cr 2\cr} \right)$, so $u_1$ and both versions of $u_2$ span the same vector space. Any nonzero ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/41349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Avoiding matching first digit of $a^n$ with $b^n$ For any given pairs of positive integers $a$ and $b$, is it possible that the first digit of $a^n$ never matches the first digit of $b^n$ for any positive integer $n$?
(If $a=2$ and $b=5$ the only possible matching first digit is "3".)
Edit: I meant to say $a=2$ in the... | I'm not answering the question, but justifying the claim in the last sentence of the question: when $a=2$ and $b=5$ the only shared leading digit of $a^n$ and $b^n$ is 3.
Suppose $2^n$ and $5^n$ share leading digit $d$ and have $r+1$ and $s+1$ digits, resp. For $n>3$ we have $d10^r < 2^n < (d+1)10^r$ and $d10^s < 5^n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/42420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)}$ On a lark (as a followup to this question), I was playing around with Wolfram alpha, and it seems that
$$\int_{0}^{1} \frac{dx}{1+{}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right)} = \frac{\log\left(\frac{2n}{2n... | First note that $${}_2F_{1}\left(\frac{1}{n},x;\frac{1}{n};\frac{1}{n}\right) = 1 + \frac{x}{n} + \frac{x(x+1)}{2! n^2} + \frac{x(x+1)(x+2)}{3! n^3} + \frac{x(x+1)(x+2)(x+3)}{4! n^4} + \cdots$$
$$ = 1 + (-x) \frac{-1}{n} + \frac{(-x)(-x-1)}{2!} \left( \frac{-1}{n} \right)^2 + \frac{(-x)(-x-1)(-x-2)}{3!} \left( \frac{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/43413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
How to find remainder modulo $n$, when $n$ is a large number I am doing RSA questions and I really could use help! Can someone show me a simple way to find $25^9 \pmod{33}$?
| For variety, let's invoke the Chinese remainder theorem
$$ 25^9 \equiv 1^9 \equiv 1 \pmod 3 $$
$$ 25^9 \equiv 3^9 \pmod{11}$$
That latter one isn't so bad to compute by repeated squaring:
$$ 3^9 \equiv 3 \cdot 3^8 \equiv 3 \cdot 9^4 \equiv 3 \cdot 81^2 \equiv 3 \cdot 4^2 \equiv 3 \cdot 5 \equiv 4 \pmod{11}$$
but I'm ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/44533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Anyone know if this integral has an analytic solution? I've come across the following integral:
$$\int_{-\pi}^{\pi}\left[\frac{1}{A-R \cos(2\theta-\phi)}\right]^{\frac{N-1}{2}}d\theta$$
I know how to approximate this integral using the Laplace method, just wondering if:
a) Does this integral have an exact answer?
b) Is... | From Maple... $K$ and $E$ are elliptic integrals.
$$
\int_{0}^{\pi} \sqrt{\frac{1}{2 - \operatorname{cos} (2 t)}} d t = \frac{2 \sqrt{3} K \Bigl(\frac{\sqrt{3} \sqrt{2}}{3}\Bigr)}{3}
$$ $$
\int_{0}^{\pi} \frac{1}{2 - \operatorname{cos} (2 t)} d t = \frac{\pi \sqrt{3}}{3}
$$ $$
\int_{0}^{\pi} \frac{1}{2 - \operator... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/45275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Inequality: $(x + y + z)^3 \geq 27 xyz$ Edit: $a,b,c$ and $x,y,z$ are positive, real numbers.
Since $(a-b)^2 \geq 0~$, $a^2 + b^2 - 2ab\geq0~$ and $a^2 + b^2 \geq 2ab~$. Similarly, $a^2 + c^2 \geq 2ac~$ and $b^2 + c^2 \geq 2bc~$.
Adding these inequalities together, $2(a^2 +b^2 + c^2) \geq 2(ab + ac +bc)~$ and accord... | I know a nice proof. It goes like this:
Let $x,y,z>0$. You know that $\frac{x+y}{2} \geq \sqrt{xy}$. This can be generalized for four numbers
$$\frac{a+b+c+d}{4}=\frac{\frac{a+b}{2}+\frac{c+d}{2}}{2}\geq \sqrt[4]{abcd}.$$
Now pick $a=x,b=y,c=z,d=\sqrt[3]{xyz}$ and you'll get your inequality.
For $x,y,z$ not positive th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/48621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 8,
"answer_id": 2
} |
Determinant of a special matrix The question is related to the eigenvalue problem. Using MAPLE, I calculated the following determinants:
$$\begin{align}
\begin{vmatrix}
-\lambda & 1 \\
1 & -\lambda \\
\end{vmatrix}&=\lambda^2-1\\
\begin{vmatrix}
-\lambda & 1 & 0 \\
1& -\lambda & 1 \\
0& 1 & -\lambd... | Based on the given results, you should be able to guess what the eigenvalues are. Can you write down an explicit set of eigenvectors for those eigenvalues? Does your construction generalize?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/54109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Which integers are the areas of squares with vertices in the 3d integer lattice? For which integers $n$ does there exist a square of area $n$ with vertices in the 3d integer lattice $\mathbb{Z}^3$?
A sufficient condition is that $n$ is a square or the sum of two squares, and I have verified that the condition is also ... | With Qiaochu's observation, one can find a general parametric solution to this problem, starting from:
$a^2 + b^2 + c^2 = d^2 + e^2 + F^2$
$a d + b e + c F = 0$
Start by forgetting about integers for now, and dividing both equations by $c^2$, and considering $a, b, d, e, F$ as rational. In other words, in effect take $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
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I need explanation for this solution for the proof. (Perfect square ends with 0,1,4,5,6,9) Give a proof to the sentence: "The final decimal digit of a perfect square is 0, 1, 4, 5, 6 or 9."
Solution:
A integer $n$ can be expressed as $10a+b$, where $a$ and $b$ are positive integers and $b$ is 0, 1, 2, 3, 4, 5, 6, 7, 8,... | Well, it should say "note that the final decimal digit of $b^{2}$ is the same as the final decimal digit of $(10-b)^{2} = 100-20b+b^{2}$." Since $100-20b+b^{2}=10(10-2b)+b^{2}$, the only number affecting the units digit of $(10-b)^{2}$ is $b^{2}$. Thus, the units digit of $b^{2}$ is the same as the units digit of $(10-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/54946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing $\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}$ How can one show :
$$\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}?$$
| HINT: The number $\alpha=\sqrt{65+18\sqrt{13}}$ is one of the four solutions of the equation $(\alpha^2-65)^2-4212=0$. Then, show that the LHS is also a solution of the same equation. The other three solutions must be disregarded somehow.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/55120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 3,
"answer_id": 2
} |
Get number of elements of a square matrix given a vector that has upper right elements of that matrix Suppose I have some vectors:
*
*$[1,2,1]$ its length is $3$ wich represents a matrix like $\begin{pmatrix}
1 & 2\\
-1 & 1
\end{pmatrix}$ the complete matrix would have $4$ elements
*$[1,2,1,3,4,1]$ its len... | Notice that all of your matrices are $n^2$ in size, where $n$ is the number of rows (or columns), and all of your vectors are 1+2+3+...+n = $n(n+1)/2$ in size. Your function takes numbers $n(n+1)/2$ to $n^2$, where $n$ is an integer. If you want to know what $n$ is from the length of your vector, just solve the quadrat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/55859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Complex number + trigo : $-1 + \tan(3)i$ , find modulus and argument I have $-1 + \tan(3)i$ and must find its modulus and its argument. I tried to solve it by myself for hours, and then I looked at the answer, but I am still confused with a part of the solution.
Here is the provided solution:
$$\begin{align}
z &= -1 +... | Let $z = -1 + \tan(3) \ i$. In the complex plane, this would be the point $(-1, \tan(3))$, which has length
$$|z| = \sqrt{(-1)^2 + \tan^2(3)} = \sqrt{1 + \frac{\sin^2(3)}{\cos^2(3)}} = \sqrt{\frac{1}{\cos^2(3)}} \sqrt{\cos^2(3) + \sin^2(3)} = \frac{1}{|\cos(3)|}$$
For the last equality, we used $\sin^2(x) + \cos^2(x) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/59315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simplify using the Tangent Difference identity I solved a problem to the point that I know the answer is
$$\frac{2nr}{\tan\left( \frac{(n-2)\pi}{2n} \right)}$$
The question tells me that the answer is going to be
$$2rn\tan\left(\frac{\pi}{n}\right)$$
Wolfram Alpha tells me the two are equal, so at least I'm on the ri... | First, as you note:
$$\tan\left(\frac{(n-2)\pi}{2n}\right) = \tan\left(\frac{\pi}{2}-\frac{\pi}{n}\right) =- \tan\left(\frac{\pi}{n}-\frac{\pi}{2}\right).$$
Now remember that $\sin\left(a - \frac{\pi}{2}\right) = -\cos(a)$ and $\cos\left(a - \frac{\pi}{2}\right) = \sin(a)$. So:
$$\begin{align*}
\tan\left(\frac{(n-2)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Triples of Numbers I have a question:
How many triples $(a,b,c)$ are there such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 1$$ and $a <b<c$? They have to be positive integers. Also find those triples.
I know that all of them have to be $\geq 2$. So do I just fix a number and count the other pairs?
If I choose $a = 3$... | Well, $\frac{1}{3} + \frac{1}{4} + \frac{1}{5} < 1$, so we must have $a=2$. So we really just need $\frac{1}{b} + \frac{1}{c} = \frac{1}{2}$.
Since $\frac{1}{4} + \frac{1}{5} < \frac{1}{2}$, $b = 3$. That leaves $c = 6$.
I think a nice way to think about it is to view the number $1$ as $\frac{1}{1}$. We can decompose $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
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finding the coefficient of $x^{14}$ in the expression: $\frac{5x^2-x^4}{(1-x)^3}$ I have a homework question which requires me to find the coefficient of $x^{14}$ in the expression: $\dfrac{5x^2-x^4}{(1-x)^3}$
I have not figured out a way to do this (I believe this is because my algebra is weak).
This is a question in ... | You are after:
\begin{align}
[x^{14}] \frac{5 x^2 - x^4}{(1 - x)^3}
&= 5 [x^{12}] (1 - x)^{-3} - [x^{10}] (1 - x)^{-3} \\
&= 5 (-1)^{12} \binom{-3}{12} - (-1)^{10} \binom{-3}{10} \\
&= 5 \binom{12 + 3 - 1}{3 - 1} - \binom{10 + 3 - 1}{3 - 1} \\
&= 389
\end{align}
Here I used:
$$
\binom{-n}{k} = (-1)^k \binom{k +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
} |
Orthogonal projection of a point onto a line How would I go about solving the following problem?
Find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$.
| The slope of the line $r$, with equation $\frac{x}{3}+\frac{y}{-5}=1$, is $m_{r}=\frac{5}{3}$ (because $\frac{x}{3}+\frac{y}{-5}=1$ is equivalent to $y=\frac{5}{3}x-5$). The slope of the line $s$ orthogonal to $r$ is $m_{s}=-\frac{3}{5}$ (because $m_{r}m_{s}=-1$). Hence the equation of $s$ is of the form
$$y=-\frac{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/62633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 5,
"answer_id": 2
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Evaluating the integral $I(u,v,w)=\iint_{(0,\infty)^2}\sinh(upq) e^{-vq^2 - wp^2}pq(q^2-p^2)^{-1}dpdq$ I found in an article "Imperfect Bose Gas with Hard-Sphere Interaction", Phys. Rev. 105, 776–784 (1957) the following integral, but I don't know how to solve it. Any hints?
$$\int_0^\infty {\int_0^\infty {\mathrm dp\... | I will assume that $u, v, w > 0$ and $4vw > u^2$. Let $$
I := \int_{0}^{\infty} \int_{0}^{\infty} \frac{\sinh (upq)}{p^2 - q^2} \, pq \, e^{-vp^2} e^{-wq^2} \; dpdq.
$$ By polar coordinate transform, we obtain $$
\begin{eqnarray*}
I & = & \int_{0}^{\frac{\pi}{2}} \int_{0}^{\infty} \frac{\sinh (r^2 u \cos \theta \si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/63534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 3,
"answer_id": 1
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How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$? How can the following inequation be proven?
$$a^2 + b^2 + c^2 \ge ab + bc + ca$$
| $a^2+b^2+c^2-ab-bc-ca\\
=(a, b, c)\begin{pmatrix}1&-1/2&-1/2\\-1/2& 1&-1/2\\-1/2&-1/2&1\end{pmatrix}\left(\begin{array}{c}a\\b\\c\end{array}\right)$
$=(a, b, c)A\begin{pmatrix}a\\b\\c\end{pmatrix}$
It is sufficient to prove $A$ is a positive semi-definite.
It follows from that the minor determinant of $A$ is one of $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/64868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 12,
"answer_id": 6
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Evaluating $ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $ I would like to evaluate: $$ \int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}\mathrm dx $$
$$ \frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}=\frac{\sqrt{1-x}+\sqrt{1+x}-2}{2(\sqrt{1-x^2}-1)} $$
The substitution $ x \rightarrow \sin(x) $ or $ \cos(x) $ can only simplify the d... | $$\newcommand{\ct}[0]{\color{grey}{\text{constant}}}
\newcommand{\b}[1]{\left(#1\right)}
\begin{align}
\int {\frac{1}{2+\sqrt{1-x}+\sqrt{1+x}}}dx&=\int\frac{-2\sin(2t)}{2+\sqrt2\sin t+\sqrt2\cos t}dt\tag{1}\\
&=-2\int\frac{\sin(2t)}{2+2\sin\b{t+\frac{\pi}4}}dt\\
&=-2\int\frac{\sin(2t)}{2+2\cos\b{\frac{\pi}4-t}}dt\\
&=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
} |
A question about limit $\lim_{n\to\infty} {n \choose n/2}/2^n$ My question is: What is the result of this limit:
$\displaystyle \lim_{n \to +\infty} \frac{{n \choose n/2}}{2^n}=$ ?
| As $\lim_{n\to+\infty} \frac{1}{2^n} \binom{n}{n/2} = \lim_{n\to+\infty} \frac{1}{4^n} \binom{2n}{n}$, and since for $r(n) = \frac{1}{4^n} \binom{2n}{n}$ the ratio
$$
r(n) >0 \land \frac{r(n+1)}{r(n)} = \frac{2n+1}{2(n+1)} < 1 \qquad \forall n \ge 1
$$
it follows that $\lim_{n \to \infty} r(n) = 0$ since $r(n)$ mono... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
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A 10 digit positive number and ordered triplets A 10 digit positive number is said to be a “LearnHub”number if its digits are all distinct and it is a multiple of 11111. How many “LearnHub” numbers are there?
Find the number of ordered triplets (a, b, c) of positive integers for which LCM (a, b) =1000, LCM (b, c) = 200... | $1000 = 2^3 * 5^3$
$2000 = 2^4 * 5^3$
Since LCM of $a$ and $b$ is $1000$, they at the most contain $2$ three times and $5$ three times.
Also the LCM of terms containing $C$ is $2000$, the term $c$ should contain $2$ four times, as $a$ and $b$ can contain $2$ only three times.
so $c$ should either be
$2^4 = 16$
$2^4 * 5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral solutions of $x^2+y^2+1=z^2$ I am interested in integral solutions of $$x^2+y^2+1=z^2.$$ Is there a complete theory comparable to the one for $x^2+y^2=z^2?$
| For the equation: $qX^2+Y^2=Z^2+j$
In the case where a square: $a=\sqrt{\frac{j}{q}}$
Using equation Pell: $p^2-(q+1)s^2=1$
Then the solution can be written:
$X=2s(s\pm{p})L\pm{ap^2}+2aps\pm{a(q+1)s^2}=bL+af$
$Y=(p^2\pm2ps+(1-q)s^2)L\pm{ap^2}+2aps\pm{a(q+1)s^2}=cL+af$
$Z=(p^2\pm2ps+(q+1)s^2)L\pm{ap^2}+2a(q+1)ps\pm{a(q+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/74931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 5,
"answer_id": 4
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Need help finding: $\frac{d}{dx}\frac{\sec{x}}{1+\tan{x}}$ So I am trying to find:
$$\frac{d}{dx}\frac{\sec{x}}{1+\tan{x}}$$
And tried doing:
$$\frac{(1+\tan{x})(\tan{x}\times\sec{x})-(\sec^{2}{x})(\sec{x})}{(1+\tan{x})^{2}}$$
Because of the Quotient Rule. Then I did some simplifying:
$$\frac{(1+\tan{x})(\tan{x}\times\... | Decided to use product rule:
$$\frac{d}{dx}\frac{\sec{x}}{(1+\tan{x})}$$
$$\frac{(1+\tan{x})(\tan{x}\times\sec{x})-(\sec^3{x})}{(1+\tan{x})^{2}}$$
I just worked on the top for a while:
$$(1+\frac{\sin{x}}{\cos{x}})(\frac{\sin{x}}{\cos{x}}\times\frac{1}{\cos{x}})-(\frac{1}{\cos{x}^{3}})$$
$$(\frac{\sin{x}}{\cos{x}}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/75292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Interscholastic Mathematics League Senior B #12 Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$.
So here is what I have done so far. I got two of the roots to be zero and 4 since $x^3(x+4)+(6x^2+1004x+1001)=0$. But for the other two roots by using the quadratic formula I get $$\fra... | We are solving the equation
$$x^4+4x^3+6x^2+1004x+1001=0.$$
Note that
$$(x+1)^4=x^4+4x^3+6x^2+4x+1.$$
Interesting! So we are looking at
$$(x+1)^4 +1000x+1000=0.$$
Let $y=x+1$. Our equation can be rewritten as $y^4+1000y=0$. The real roots are $y=0$ and $y=-10$, and the other two roots are non-real. Now it's essential... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Formula for completing the square? My math teacher said that this was the formula for completing the square.
Original function: $$ax^2 + bx + c$$
Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$
However, using this formula I'm not getting the same answers that I would get just by determining ... | For the polynomial $4x^2+4x+5$ that you mention, I would not use the formula, since it is fairly clear that $4x^2 +4x$ is "almost" $(2x+1)^2$. In fact, $(2x+1)^2=4x^2+4x+1$, so $4x^2+4x+5=(2x+1)^2-1+5=(2x+1)^2+4$.
In general, suppose that $a \ne 0$, and we want to deal with $ax^2+bx+c$. Multiply the expression by $4a$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/76772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Irreducible polynomial which is reducible modulo every prime
How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$?
For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not solutions of $x^4+1=0$ then if it is reducible the factors are ... | For every odd prime $p$ we have $8\mid p^2-1$. The multiplicative group of the finite field $F=GF(p^2)$ is cyclic of order $p^2-1$. Putting these two bits together tells us that there is a primitive root $u$ of order $8$ in $F$. We must have $u^4=-1$, because $-1$ is the only element of multiplicative order two. Becaus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
"answer_count": 3,
"answer_id": 1
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Solving the equation $\frac{2x+3}{x+1} = \frac{2x+2}{x-1}$ I'm a college student and I've got a problem solving this:
$$\frac{2x+3}{x+1} = \frac{2x+2}{x-1}.$$
Since neither the numerators nor the denominators are equal, I figured that $(2x+3)(x-1) = (x+1)(2x+2)$ would provide me with a suitable $x$.
However calculating... | There’s nothing wrong with your basic approach, but you seem to have solved the equation $(2x+3)(x-1)=(x+1)(2x+2)$ incorrectly. After multiplying out both sides, you should have $$2x^2+x-3=2x^2+4x+2\;,$$ which simplifies progressively to $$\begin{align*}
x-3&=4x+2\;,\\
-3&=3x+2\;,\\
-5&=3x\;,\text{ and}\\
x&=-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Summation Identity: $\sum_{i=1}^ni^3 = \left( \frac{n(n+1)}{2} \right)^2$ I have to prove:
$$\sum\limits_{i = 1}^n i^3 = \Bigg( \frac{n(n+1)}{2}\Bigg)^2$$
Using the following:
$$n^3 = 6 {n \choose 3} + 6 {n \choose 2} + n \quad \forall n \in \mathbb{N}$$
My work is that first I substitute $n^3$ for $6 {n \choose 3} + 6... | Suppose we have arrived at the expression
$$6{n+1 \choose 4} + 6{n+1 \choose 3} + {n+1 \choose 2},$$
and we remain in a combinatorial mood, not an algebraic one. Then we might work a little harder and give a bijective argument.
From a group of $n+1$ boys and $n+1$ girls, we can choose $2$ boys and $2$ girls (a pair of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Show that the value of a definite integral is unity $$ \int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx=1$$
| We show that $I=\int_2^4 \frac{f(x)}{f(x)+f(6-x)}\,dx$ equals $1$, where $f(x)=\sqrt{\log(9-x)}$.
Proof. By making a substitution $y=6-x$, we get
$$I=\int_4^2 -\frac{f(6-y)}{f(y)+f(6-y)}\,dy=\int_2^4 \frac{f(6-y)}{f(y)+f(6-y)}\,dy.$$
Therefore
$$\begin{align*}2I&=\int_2^4 \frac{f(x)}{f(x)+f(6-x)}dx+\int_2^4 \frac{f(6-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/80754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Square Roots of Complex Number $3-4i$
What I did
$z^2=3-4i$
$(a+bi)^2 = 3-4i$
$a^2-b^2+2abi = 3-4i$
Then got 2 simultaneous equations
$a^2-b^2=3$ and $2ab=-4$
Solve for $a^2$ in 1st equation: $a^2=3+b^2$
Subbed into 2nd equation to power of 2
$(3+b^2)b^2=4$
$b^4+3b^2-4=0$
Is there a better way than below? Solving powe... | You can look at this as a problem in the arithmetic of the Gaussian Integers, $\mathbb Z[i]$. I’ll make use of the fact that this ring is a Unique Factorization Domain, and I’ll also use the fact that every prime $p\equiv1\pmod4$ is writable as the sum of two squares, $p=m^2+n^2$, equivalently, $p=(m+ni)(m-ni)$, and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
How to calculate the determinant of all-ones matrix minus the identity? How do I calculate the determinant of the following $n\times n$ matrices
$$\begin {bmatrix}
0 & 1 & \ldots & 1 \\
1 & 0 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & ... & 0
\end {bmatrix}$$
and the same matrix but one of columns rep... | $$D_n(a,b)=
\begin{vmatrix}
a & b & b & b \\
b & a & b & b \\
b & b & a & b \\
b & b & b & a
\end{vmatrix}$$
($n\times n$-matrix).
$$D_n(a,b)=
\begin{vmatrix}
a & b & b & b \\
b & a & b & b \\
b & b & a & b \\
b & b & b & a
\end{vmatrix}$$
$$=[a+(n-1)b]
\begin{vmatrix}
1 & 1 & 1 & 1 \\
b & a & b & b \\
b & b & a & b \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 6
} |
how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method? how to solve $\pm y \equiv 2x+1 \pmod {13}$ with Chinese remainder theorem or iterative method?
It comes from solving $x^2+x+1 \equiv 0 \pmod {13}$ (* ) and background is following:
13 is prime. (* ) holds under Euclidean l... | To solve $x^2 + x + 1 \equiv 0 \pmod{13}$, you can use the usual quadratic formula, interpreted appropriately. The solutions are given by
$$x = \frac{-1 \pm\sqrt{-3}}{2},$$
where "$\sqrt{-3}$" means an integer $y$ such that $y^2\equiv -3\pmod{13}$, and "$\frac{1}{2}$" means an integer $z$ such that $2z\equiv 1\pmod{13}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/85814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Probability of rolling a die I roll a die until it comes up $6$ and add up the numbers of spots I see.
For example, I roll $4,1,3,5,6$ and record the number $4+1+3+5+6=19$. Call this sum $S$.
Find the standard deviation of $S$.
I have been looking for an easy way to do this because I know I can use the definitions h... | Let $Y$ be the number of rolls before a $6$ is rolled. Let $X$ be the sum of the dice rolled before a $6$. Straightforward calculation yields
$$
\mathsf{E}(X|Y=n)=n\;\mathsf{E}(X|Y=1)=3n
$$
and
$$
\mathsf{Var}(X|Y=n)=n\;\mathsf{Var}(X|Y=1)=2n
$$
and
$$
\mathsf{P}(Y=n)=\left(\frac{5}{6}\right)^n\frac{1}{6}
$$
Usin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Finding a matrix How might I find matrix $M\in M_2(\mathbb C)$ such that $M^t A =M^{-1}$
where $A=\left[
\begin{array}{cc}
a & a \\
a & a+1 \\
\end{array}
\right]\in M_2(\mathbb... | Clearly $a$ is nonzero, or else $M^\top A$ is non-invertible and it cannot be equal to the invertible matrix $M^{-1}$. Now,
$$
\begin{align}
A=\begin{pmatrix}a&a\\a&a+1\end{pmatrix}
&=\begin{pmatrix}a&a\\a&a\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}\\
&=\begin{pmatrix}\sqrt{a}\\ \sqrt{a}\end{pmatrix}\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/87034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to simplify $\frac{(3x^{3/2}y^3)} {(x^2y^{-1/2})}^{-2}$? I can't figure this one out on my own either
$$\frac{(3x^{3/2}y^3)} {(x^2y^{-1/2})}^{-2}$$
I am a little confused on all the small rules at play here but I know that a negative exponent will flip a fraction so I square the top and then flip it. But before tha... | You have:
$$\begin{align*}
\frac{(3x^{3/2}y^3)^{-2}}{(x^2y^{-1/2})} &= \frac{1}{(3x^{3/2}y^3)^2}\times \frac{1}{x^2} \times y^{1/2}\\
&= \frac{1}{(3^2)(x^{3/2})^2(y^3)^2}\times \frac{1}{x^2}\times\frac{y^{1/2}}{1}\\
&= \frac{1}{9x^3y^6}\times\frac{1}{x^2}\times \frac{y^{1/2}}{1}\\
&= \frac{1}{9x^3x^2}\times\frac{y^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Systems of equations finding right triangles I need help setting up the equation for the question, "Find all right triangles for which the perimeter is $24$ units and the area is $24$ square units."
I know that the area is $A = \frac12 b h$ and perimeter is $P = a + b + h$.
Using this, would the system look something ... | You might be more comfortable using $a$ and $b$ for the two "legs" of the triangle, and $c$ for the hypotenuse.
Then, exactly as you did, we obtain
$$a+b+c=24 \qquad \text{and}\qquad \frac{1}{2}ab=24.\qquad\qquad(1)$$
By the Pythagorean Theorem, we have
$$a^2+b^2=c^2.\qquad\qquad(2)$$
Now what? One approach (but not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Geometric inequality: $2r^2+8Rr \leq \frac{a^2+b^2+c^2}{2}$ Suppose $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $R$ and $r$ are its circumradius and inradius respectively. How can one prove the following inequality? $$2r^2+8Rr \leq \frac{a^2+b^2+c^2}{2}$$
| We know that
$$\begin{align*}
a+b+c&=2s,\\
ab+bc+ca&=s^2+r^2+4rR,
\end{align*}$$
where $s$ is semiperimeter, $r$ is inradius and $R$ is circumradius;
see e.g., Manfrino, Ortega, Delgado - Inequalities: A Mathematical Olympiad Approach, Section 2.5.
Using these inequalities we can rewrite the LHS as
$$
\begin{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Conditional Normal $X,Y,Z$ are standard normal R.V.
What is the value of $\operatorname{E}[X|X+Y+Z=1]$ and $\operatorname{Var}(X|X+Y+Z=1)$?
I think the first one should be $1/3$ by symmetry but don't know how to tackle the second part.
Any thought on this? Thanks a lot!
| If $A$ and $B$ are jointly normal random variables, then the conditional density of $A$ given $B = b$ is a normal density with mean and variance as follows:
$$E[A\mid B = b] = \mu_A + \left.\left.\frac{\text{cov}(A,B)}{\sigma_B^2}
\right(b - \mu_B\right), ~~
\text{var}(A\mid B = b) = \frac{\sigma_A^2\sigma_B^2
- (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find a sum of appropriate values of $\cos$ and $\sin$ to determine the value of a series The task is to find a sum of multiple values $\cos$ and $\sin$ to determine the value of
$$\sum_{n=1}^\infty (-1)^n \frac{n}{(2n+2)!}$$
Since I had no clue how to approach this I consulted Wolfram|Alpha which returned this result:... | Of course you know the series for $\cos(1)$ and $\sin(1)$, so you want to express this
using those.
I prefer to start the sum at $n=0$: since the $n=0$ term is $0$, this is harmless.
Note that $\frac{n}{(2n+2)!} = \frac{n}{(2n+1)!(2n+2)}$. Now
$\frac{n}{2n+2} = \frac{1}{2} - \frac{1}{2n+2}$. So
$$\sum_{n=0}^\infty (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
How does one prove $\int_0^\infty \prod_{k=1}^\infty \operatorname{\rm sinc}\left( \frac{t}{2^{k+1}} \right) \mathrm{d} t = 2 \pi$ Looking into the distribution of a Fabius random variable:
$$
X := \sum_{k=1}^\infty 2^{-k} u_k
$$
where $u_k$ are i.i.d. uniform variables on a unit interval, I encountered the follo... | Since $u_k$ and $1-u_k$ are equal in distribution, it follows that
$$
X \stackrel{d}{=} \sum_{k=1}^\infty 2^{-k} \left(1-u_k\right) = \sum_{k=1}^\infty 2^{-k} - X = 1 - X
$$
Since $u_k \geq 0$, it follows that $\mathbb{P}\left(0 \leqslant X \leqslant 1\right) = 1$.
Therefore $f_X(x) = 0$ for $x < 0$ or $x>1$. It ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
} |
For what values of $b$ does this function lack extrema? I need to find the range of values of the parameter $b$ for which the function below has no extrema.
$$ \frac{b}{3}\,8^x + (2) 4^x + (b+3) 2^x + b \ln(2)$$
In the beginning I thought it'd be easy, Very simple, just finding a range when derivative is not zero.... | Let $f(b,x) = \frac{b}{3}8^x + 2 \cdot 4^x + (b+3) 2^x + b \log(2)$. What you want is the set of values for $b$ such that $\frac{\partial}{\partial x}f(b,x)\neq 0$ for all $x\in\mathbb{R}$. Computing the partial derivative with respect to $x$ gives
$\begin{eqnarray*}
\frac{\partial}{\partial x}f(b,x) &=& \frac{\partia... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/97977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Given a die, what is the probability that the second roll of a die will be less than the first roll? If you are given a die and asked to roll it twice. What is the probability that the value of the second roll will be less than the value of the first roll?
| If the:
first roll is a 6 odds are: 5/6
first roll is a 5 odds are: 4/6
first roll is a 4 odds are: 3/6
first roll is a 3 odds are: 2/6
first roll is a 2 odds are: 1/6
first roll is a 1 odds are: 0/6
Therefore the total odds are the average of all those roll possibilities so:
$$
\frac{\frac{5}{6} + \frac{4}{6} + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "97",
"answer_count": 10,
"answer_id": 4
} |
Which is the fastest way to find the remainder when $2^{400}$ is divided by $400$?
Which is the fastest way to find the remainder when $2^{400}$ is divided by $400$?
My approach is to break up $400$ as $16 \times 25$ and then apply CRT, I was wondering if there is any other approach that gives the result faster than... | As a relatively fast but purely mechanical method (not a whole lot of thought involved), I'd first note that the exponent $$400_{10}=110010000_2=2^4+2^7+2^8$$ so that $$2^{400}=2^{(2^4+2^7+2^8)}=2^{2^4}\cdot2^{2^7}\cdot2^{2^8}.$$
Now, by successively squaring:
$$\begin{align}
2^{2^1}&\equiv 4\mod 400
\\
2^{2^2}\equi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/100982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
What's the prime period of $\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$ What's the prime period of the following function?
$$\frac{\sin 2x + \cos 2x}{\sin 2x - \cos 2x}$$
| $$\begin{align*}
\frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}&=\frac{\sin 2x+\cos 2x}{\sin 2x-\cos 2x}\cdot\frac{\sin 2x+\cos 2x}{\sin 2x+\cos 2x}\\\\
&=\frac{(\sin 2x+\cos 2x)^2}{\sin^2 2x-\cos^2 2x}\\\\
&=-\frac{\sin^2 2x+2\sin 2x\cos 2x+\cos^2 2x}{\cos 4x}\\\\
&=-\frac{1+\sin 4x}{\cos 4x}\\\\
&=-\sec 4x-\tan 4x\;.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/101581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Explanation of the Fibonacci sequence appearing in the result of 1 divided by 89? What's the explanation of the Fibonacci sequence appearing in the result of 1/89, as demonstrated by http://www.goldennumber.net/Number89.htm and shown below? If you wish, also explain the relation to the number 109 too.
1 / 89 =
0 / (10 ... | Consider the generating function
$$f(x) = \sum_{k=0}^{\infty} F_k x^k$$
where $F_k$ are the Fibonacci numbers. The Fibonacci recurrence $F_k = F_{k-1} + F_{k-2}$ gives
$$f(x) = x + \sum_{k=2}^{\infty} (F_{k-1} + F_{k-2}) x^k = x + \sum_{k=1}^{\infty} F_k x^{k+1} + \sum_{k=0}^{\infty} F_k x^{k+2} = x + (x + x^2) f(x).$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
$\frac{16}{27} \left( \frac a{b+c} + \frac b{a+c} +\frac c{a+b} \right) ^3 + \left( \frac{abc}{(a+b)(b+c)(a+c)}\right)^{\frac{1}{3}} \geq \frac52$ I don't quite remember where this problem is from. I came across is sometime last summer, when I was in an olympiad-problem mood and I decided to improve my inequality skill... | First, make the substitutions
$$
x=
\frac{a}{b+c},
\quad
y=
\frac{b}{a+c},
\quad
z=
\frac{c}{a+b}.
$$
The strategy will be to reduce the problem to an inequality in the single variable $t=(xyz)^{1/3}$. Note that $xy+yz+xz+2xyz=1$, and the inequality to be proved is
$$
\frac{16}{27}\left(x+y+z\right)^3+(xyz)^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/104176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
"answer_id": 0
} |
Perfect Squares ending in 576 I want to find out perfect squares ending in 576, after the number 576.
Here is my derivation to arrive at such a number. Let the perfect square ending in $576$ be $1000k+576$. Every perfect square can be expressed as a the sum of a certain number of consecutive odd numbers. For eg: $2^2 =... | The flaw is
Since, (1+3+5+7+ ....49) = 576
The sum is actually 625. You need to stop a term earlier.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/104277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$ I know that $\mathbb Z[\sqrt{7}]$ is a UFD, and I can write the equation $(2 + \sqrt{7})(3 - 2\sqrt{7}) = (5 - 2\sqrt{7})(18 + 7\sqrt{7}) $. So clearly these are not all irreducibles. How do I determine which of these aren't irreducible? How d... | An irreducible of $\mathbb{Z}[\sqrt{7}]$ must be a prime of $\mathbb{Z}$, or must lie above a prime of $\mathbb{Z}$.
Since $\mathbb{Z}[\sqrt{7}]$ is the splitting field of $x^2-7$, you want to determine how $x^2-7$ factors modulo $p$, for $p$ prime. In other words, for which primes is $7$ a square modulo $p$?
If $p=2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Correlation between variables I asked this question on stats SE but did not find a suitable answer so far. Maybe someone can help.
Given n random variables x1,...,xn (one-dimensional).
The following is known (corr() = Pearson correlation):
corr(x1,x2) = a
corr(x2,x3) = a
The actual values of the random variables and... | I find it most intuitive to use the cholesky-decomposition of some correlation-matrix to look at such questions. The cholesky-decomposition provides a lower triangular matrix which always has (given the variables $\small x_1,x_2,x_3 $) the form
$\qquad \small \begin{array} {r|lll}
x_1: & 1 & . & . & \\
x_2: & a_1 & a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/106101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
$f(x) = x^\frac{1}{3}$ is continuous at any $p \neq 0$. Trying to show $f(x) = x^\frac{1}{3}$ is continuous at any $p \neq 0$. The case with 0 is easy, this one has me stuck. I should use that $a^3 - b^3 = (a-b)(a^2+ab+b^2).$ I started off by $$|x^\frac{1}{3} - p^\frac{1}{3}| = |x^\frac{1}{3} - p^\frac{1}{3}| \frac{|x^... | If you make sure that $|x-p|<|p|$ (as part of how you choose $\delta$), then $x$ and $p$ have the same sign, and all terms in the denominator are positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/107732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How can I make the following 2 fractions integers? Let $m,n$ be integers. I want to find the possible values of $m,n$ such that $4(m+n)\over (2m+n)^2+3n^2$ and $4n\over (2m+n)^2+3n^2$ are both integers too. Would someone please help? Of course letting $(2m+n)^2+3n^2=4$ gives some good values, but is this all the $m,n$ ... | We find all integer values of $m$ and $n$ such that $\frac{4n}{(2m+n)^2+3n^2}$ is an integer. Once this is done, your problem that imposes additional conditions is easily solved.
If $3n^2 > 4n$, then the bottom has absolute value greater than the absolute value of the top. Thus all but $n=0$, $n=\pm 1$ are immediate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Telescoping Series I have a question about a particular formula that is supposed to be used to simplify difficult summations into telescoping series. The formula is as follows.
$$\sum k(k+1) = \sum \frac{1}{3} \Big(k(k+1)(k+2) - (k-1)k(k+1)\Big) $$
So now here is the question. How would this be used if one were to ... | We have
$$k(k+1)(k+2)= \frac{1}{4}(k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)),$$
which gives a telescoping sum to yield $$\sum_{k=1}^n k(k+1)(k+2)=\frac{1}{4}n(n+1)(n+2)(n+3).$$
Similarly, from the identity you wrote we get $$\sum_{k=1}^n k(k+1)=\frac{1}{3}n(n+1)(n+2),$$ and from the identity $$k=\frac{1}{2}(k(k+1)-(k-1)k)$$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solving a differential equation by power series. Here's a question I'm struggling with and I'd like help on it. Please, this is not a homework problem.
I want to find the power series solutions about the origin of two linearly independent solutions of $$w''-zw=0.$$
Also, how do I show that these solutions are analy... | $w(x)=a_0+a_1x+\frac{a_2x^{2}}{2!}+\frac{a_3x^{3}}{3!}+...$
$w''(x)=a_2+a_3x+\frac{a_4x^{2}}{2!}+\frac{a_5x^{3}}{3!}+...$
$w''-xw=0$
$(a_2+a_3x+\frac{a_4x^{2}}{2!}+\frac{a_5x^{3}}{3!}+...)-x(a_0+a_1x+\frac{a_2x^{2}}{2!}+\frac{a_3x^{3}}{3!}+...)=0$
$a_2+(a_3-a_0)x+(\frac{a_4}{2!}-a_1)x^{2}+(\frac{a_5}{3!}-\frac{a_2}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/109899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Easy way to solve this non-linear second degree DY $\sqrt{y}\;y''=1$? I prooceeded by integrating both sides $$y'=\int y^{-\frac{1}{2}} dx=\cdots$$
so I got $(y')^{2}+C y' - \frac{1}{2} \sqrt{y} = 0$ but I am thinking that I am proceeding the wrong or the hard way. Some easy to solve this kind of 2nd-degree DYs?
Page 6... | $$y'=\int y^{-\frac{1}{2}} dx=...$$
$$y''= y^{-\frac{1}{2}} $$
$$y'y''= y'y^{-\frac{1}{2}} $$
$$\int y'y'' dx=\int y^{-\frac{1}{2}} y'dx$$
$$\frac {y'^{2}}{2} = 2y^{\frac{1}{2}} +k $$
$$y'^{2} = 4y^{\frac{1}{2}} +2k=4y^{\frac{1}{2}} +c $$
$$y' = \sqrt{4y^{\frac{1}{2}} +c} $$
$$\frac{y'}{\sqrt{4y^{\frac{1}{2}} +c}} = 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
In how many ways can a batsman score a century in all $6$'s and $4$'s? A batsman scored a century in all $6$'s and $4$'s. In how many ways can he do this?
The given answer is $8$, but there is no explanation, how are they doing it?
As century is $100$ runs a very loose translation of this problem would be "In how many ... | If you want exactly $100$ (and not $102$ or $104$, etc)
Then these are the possibilities
$10\cdot 6+10\cdot 4=100$,
$12\cdot 6+7\cdot 4=100$,
$14\cdot 6+4\cdot 4=100$,
$16\cdot 6+1\cdot 4=100$,
$8\cdot 6+13\cdot 4=100$,
$6\cdot 6+16\cdot 4=100$,
$4\cdot 6+19\cdot 4=100$,
$2\cdot 6+22\cdot 4=100$
And if you allow that n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/110579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
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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.