All Questions
5
questions
51
votes
9
answers
6k
views
Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
Question : Is the following true for any $m\in\mathbb N$?
$$\begin{align}\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}\qquad(\star)\end{align}$$
Motivation : I reached $(\star)$ by ...
- 145k
42
votes
3
answers
1k
views
Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
- 13.2k
8
votes
2
answers
685
views
Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$
I want to show that
$$\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})} = \frac{n^2-1}6$$
With induction I don't know how I could come back from $\frac{1}{1-\cos(\frac{2k\pi}{n+1})}$ to $\frac{1}{1-\...
- 81
3
votes
1
answer
262
views
On $\mathrm{\sum_{x\in\Bbb Z}sech(x)=3.142242…}$
Inspired by
This question,
I started to wonder about simpler series. I have seen similar questions to the following, but none had this special case explicitly. It is related to the q-digamma ...
- 12.5k
12
votes
1
answer
267
views
Could this conjecture be proved ? (sum of even powers of cotangents in arithmetic progression )
Having tried (in vain) to answer this question, I worked the explicit formulae of
$$\color{blue}{S_k=\sum _{n=1}^m \Big[\cot \left(\frac{n \,\pi }{2 m+1}\right)\Big]^k}$$ where $k$ is an even integer....
- 268k