All Questions
27
questions
4
votes
0
answers
87
views
How deduce $\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ from $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$?
I know that $\displaystyle\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ for any integer $n \geq 1$ is true.
Now, suppose that $n$ is odd, how show
$$
\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\...
- 323
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
1
vote
1
answer
120
views
Find the value of sum $\forall\:\:\alpha,\beta\in\mathbb{R}$
Evaluate the sum $\forall\:\:\alpha,\beta\in\mathbb{R}$ $$S=\sum_{n=1}^{\infty}\frac{\alpha^{n+1}-1}{n(n+1)}\sin\left(\frac{n\pi}{\beta}\right)$$
I rewrote this as $$S=\sum_{n=1}^{\infty}\left(\left(\...
3
votes
0
answers
107
views
Series representation of $n$th derivative of $x^n/(1+x^2)$
Find the nth derivative of $\frac{x^n}{1+x^2}$.
Please I need help in this. They are further asking to show that when $x=\cot y$ the nth derivative can be expressed as
$$n!\sin y\sum_{r=0}^{n}(-1)^r {...
2
votes
0
answers
97
views
Integral of $\,\tan^{2n}(x)\,\mathrm dx$
I want to evaluate $\,\displaystyle I_{n}=\int_{0}^{\frac{\pi}{4}} \tan^{2n}(x)\,\mathrm dx$.
I proved that $\,I_{n}+I_{n-1}=\dfrac{1}{2n-1}\,,\,$ where $I_{0}=\dfrac{\pi}{4}$.
From that I found that (...
user1009942
1
vote
2
answers
66
views
Closed form for $\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x)$
I request a closed form for the following sum $$S(n)=\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x) $$
I tried using De Moivre's theorem $$\cos(nx)+i\sin(nx)=(\cos (x)+i\sin(...
user1103841
0
votes
0
answers
45
views
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$ [duplicate]
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$.
By Euler's theorem, $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$ for every real number x. Also, I know that $\sum_{n=1}^\infty \dfrac{1}{n^2} = \dfrac{\pi^2}6.$ ...
- 1,837
0
votes
0
answers
95
views
Proof of $sin$ formula.
I am reading this quesiton and accepted answer.
Question is about proof.
$S = \sin{(a)} + \sin{(a+d)} + \cdots + \sin{(a+nd)}$
$S \times \sin\Bigl(\frac{d}{2}\Bigr) = \sin{(a)}\sin\Bigl(\frac{d}{2}\...
- 879
0
votes
2
answers
80
views
Proof explanation for a sum of sines [duplicate]
Looking through proofs and there's one part I'm confused about, that if solved will get me straight to the answer.
I need to show that $\sum_{m=0}^{N-1} \sin((m+\frac{1}{2})x) = \frac{\sin(Nx/2)^2}{\...
- 95
1
vote
1
answer
133
views
Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $
Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $ where $f(x)= sin(log_e x)(\frac{1}{x^{a}}-\frac{1}{x^{1-a}})$,$ 0<a<1/2$
My try -
$\lim_{N\to +\infty}[ \int_{1}...
user826941
4
votes
1
answer
288
views
Evaluating $\sum_{k=0}^{2020} \cos( \frac{2πk}{2021}$)
I am trying to come up with a solution to $$\sum_{k=0}^{2020}\cos\left(\frac{2\pi k}{2021}\right) $$ so far I have proceeded as to acknowledge the cosine is just the real part of the Euler's formula ...
- 41
5
votes
2
answers
240
views
Proving $\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+1}{n+1})}k>\sum_{k=1}^n\frac{\sin(k\pi\frac{2m+3}{n+1})}k$
A few days ago I asked a question about an interesting property of the partial sums of the series $\sum\sin(nx)/n$.
Here's the link:
Bound the absolute value of the partial sums of $\sum \frac{\sin(nx)...
- 1,684
1
vote
0
answers
129
views
Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$
Problem: Find Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$.
This is the follow up of Prove that $|\sin 1| + |\sin 2| + |\sin 3| +\cdots+ |\sin 3n| > 8n/5$.
My attempt
By the ...
- 40.3k
17
votes
2
answers
579
views
Is there a way to evaluate analytically the following infinite double sum?
Consider the following double sum
$$
S = \sum_{n=1}^\infty \sum_{m=1}^\infty
\frac{1}{a (2n-1)^2 - b (2m-1)^2} \, ,
$$
where $a$ and $b$ are both positive real numbers given by
\begin{align}
a &= ...
10
votes
2
answers
889
views
Find $\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$
Find $$M:=\sum_{n=1}^{\infty}\tan^{-1}\frac{2}{n^2}$$
There's a solution here that uses complex numbers which I didn't understand and I was wondering if the following is also a correct method.
My ...
- 1,206