All Questions
12
questions
7
votes
2
answers
288
views
Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$$
here it can be shown that where $x=y$,
$$n^2 \csc^2(nx) = \sum_{k=0}^...
- 1,379
3
votes
0
answers
103
views
How can one find an identity for $\sum_{k=0}^{n-1}\tan^2(\frac{k\pi y}{n}) $?
Using the function
$$
f(z)=\frac{n/z}{z^n-1}\left(\frac{z-1}{z+1}\right)^2\tag{1}
$$
and residue theorem
$$
\sum_{k=0}^{n-1}\tan^2\left(\frac{k\pi}{n}\right)=n^2-n\tag{2}
$$
can be found according to ...
- 1,379
9
votes
1
answer
537
views
Is there any identity for $\sum_{k=1}^{n}\tan\left(\theta+\frac{k\pi}{\color{red} {2n+1}}\right)$?
Is there any identity for $\sum_{k=1}^{n}\tan\left(\theta+\frac{k\pi}{\color{red} {2n+1}}\right)$ or $\sum_{k=1}^{n}\tan\left(\frac{k\pi}{\color{red} {2n+1}}\right)$ ?
I thought maybe wrongly that ...
- 1,379
1
vote
1
answer
161
views
$\sum_{n=1}^{\infty} \frac{\sin(n^2)}{n^2}$
Question:
$$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{n^2}=\,?$$
Previously I calculated a similar summation but it was more luck than wisdom, and insight led me to believe my methods were super incorrect (...
- 593
4
votes
0
answers
128
views
How to solve $\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}$?
I need to solve this sum:
$$\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}.$$
Do you have any ideas for how I could do this?
I know that this sum:
$$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^...
- 1,027
1
vote
1
answer
306
views
How to solve sum of cos(kx) for the case cos(x)=1
I have the solution for $\sum_{k=1}^n \cos(kx)$:
\begin{align}
\sum_{k=1}^n \cos(kx) & = \Re\left(\sum_{k=1}^n e^{ikx}\right)\\
& = \Re\left(e^{ix} {e^{inx}-1 \over e^{ix}-1}\right) \\
& ...
- 9
-1
votes
1
answer
92
views
How to prove $\sum_{n=1}^{2k} (-1)^n\sin\frac{n^2\pi}{4k}=(-1)^k\sqrt{k/2}$ [closed]
$$\sum_{n=1}^{2k} (-1)^n\sin\frac{n^2\pi}{4k}=(-1)^k\sqrt{k/2}$$
How to prove it without induction? Any helps would be appreciated.
0
votes
3
answers
202
views
Showing $1 + 4\cos{2\theta} + 6 \cos{4\theta} +4\cos{6\theta}+ \cos {8\theta} =16\cos{4\theta} \cos^4 \theta$.
Show that $1 + 4\cos{2\theta} + 6 \cos{4\theta} +4\cos{6\theta}+ \cos {8\theta} = 16\cos{4\theta} \cos^4 \theta$
Do I solve this question by using summation of series?
2
votes
1
answer
322
views
Sum of sines $\sum_{k=0}^{n} \sin(\phi +k\alpha)$
I've got the following problem. I'd like to prove that
$$\sum_{k=0}^{n} \sin(\phi +k\alpha) = \frac{\sin\left(\frac{n+1}{2}\right)\alpha + \sin\left(\phi + \frac{n\alpha}{2}\right)}{\sin\frac{\alpha}{...
- 1,557
25
votes
5
answers
2k
views
Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
In old popular science magazine for school students I've seen problem
Prove that $\quad $
$\dfrac{1}{\cos^2 20^\circ} +
\dfrac{1}{\cos^2 40^\circ} +
\dfrac{1}{\cos^2 60^\circ} +
\dfrac{1}{\cos^...
- 17.4k
21
votes
7
answers
18k
views
Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
- 23.2k
13
votes
3
answers
16k
views
How to prove Lagrange trigonometric identity [duplicate]
I would to prove that
$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+
\frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$
given that
$$1+...
- 2,318