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4
questions
1
vote
2
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computation of $\sum_{k=0}^{2p-1}( \cos(x+\frac{k \pi}{2p}))^{2p}$
Let $p \in \mathbb{N^*}, x \in \mathbb{R}$, someone I know was trying to compute the following sum:
$$\sum_{k=0}^{2p-1}\left( \cos\left(x+\frac{k \pi}{2p}\right)\right)^{2p}$$
It seems that the result ...
- 1,928
2
votes
1
answer
94
views
Prove that for integer m and N, this sum with N-1 terms of cosec raised to 2m multiplied by (N/2)^m is an integer.
Prove that for integers $m \ge 1$, $N \ge 2$,
$F(m,N)=\large \frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\frac{\pi j}{N}\right)$ is an integer.
I encountered this ...
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-1
votes
3
answers
174
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If $\sum_{n=1}^\infty\tan^{-1}\left(\frac4{n^2+n+16}\right)=\tan^{-1}\left(\frac\alpha{ 10 }\right)$, then find $\alpha$.
$$\sum _ { n = 1 } ^ { \infty } \tan ^ { -1 } \left( \frac { 4 } { n ^ {
2 } + n + 16 } \right)= \tan ^ { -1 } \left( \frac { \alpha } { 10 }
\right)$$ Find $\alpha$.
I know I need to convert to $$\...
- 245
51
votes
9
answers
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Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$
Question : Is the following true for any $m\in\mathbb N$?
$$\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 ...
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