All Questions
4
questions
9
votes
2
answers
356
views
computing $A_2=\sum_{k=1}^{n}\frac{1}{(z_k-1)^2} $ and $\sum_{k=1}^n \cot^2\left( \frac{k\pi}{n+1}\right)$
Assume that $z_1,z_2,...,z_n$ are roots of the equation $z^n+z^{n-1}+...+z+1=0$.
I was asked to compute the expressions
$$A_1=\sum_{k=1}^{n}\frac{1}{(z_k-1)} ~~~~~~and~~~~~~A_2=\sum_{k=1}^{n}\...
- 24.2k
0
votes
0
answers
226
views
Proving an identity with geometric series.
I've been at this for MANY hours and I think it's time I sought help.
Question: Given $k = \frac{2 \pi}{Na}\left ( p-\frac{N}{2} \right )$, prove that $\sum_{k=1}^{N}e^{ika\left ( n-m \right )}=N\...
- 6,363
0
votes
3
answers
2k
views
If $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + ... + \omega^{n-1} = 0 $? [duplicate]
Let $\omega = e^{(\frac{2\pi i}{n})}$ why $1+ \omega + \omega^{2} + ... + \omega^{n-1} = 0 $?
I saw this on a algebra PPT slice. However the teacher did not explain why this equation is correct, can ...
- 269
3
votes
2
answers
176
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Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$
Evaluate
$$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$
where $i=\sqrt{-1}$
For this question, I did the following,
Let
$$
\begin{align*}
S &= \sum_{n=1}^{\infty} \...
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