All Questions
9
questions
10
votes
3
answers
354
views
Finding $\sum_{k=0}^{n-1}\frac{\alpha_k}{2-\alpha_k}$, where $\alpha_k$ are the $n$-th roots of unity
The question asks to compute:
$$\sum_{k=0}^{n-1}\dfrac{\alpha_k}{2-\alpha_k}$$
where $\alpha_0, \alpha_1, \ldots, \alpha_{n-1}$ are the $n$-th roots of unity.
I started off by simplifiyng and got ...
- 7,551
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
$$A_1=\sum_{k=1}^{n}\frac{1}{(z_k-1)} ~~~~~~and~~~~~~A_2=\sum_{k=1}^{n}\...
- 24.2k
6
votes
4
answers
3k
views
if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then...
if 1, $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, $\alpha_{n-1}$ are nth roots of unity then
$$\frac{1}{1-\alpha_1} + \frac{1}{1-\alpha_2} + \frac{1}{1-\alpha_3}+\ldots+\frac{1}{1-\alpha_n} = ?$$
...
- 1,396
3
votes
4
answers
174
views
Evaluate $\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}$, where $q^7=1$ and $q\neq 1$.
Let $q$ be a complex number such that $q^7=1$ and $q\neq 1$. Evaluate $$\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}.$$
The given answer is $\frac{3}{2}$ or $-2$. But my answer is $\pm 2$.
...
- 373
1
vote
1
answer
161
views
Summation of rational complex function
Let $z$ be a non-real complex number such that $z^{23}=1$. Compute $$\sum_{k=0}^{22} \frac {1}{1+z^k +z^{2k}}$$.
I could not really do much about this problem. I tried writing the summand as a ...
- 407
56
votes
7
answers
8k
views
Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?
It might be obvious that $2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { 2\sqrt { \cdots } } } } } } $ equals $4.$ So what about $i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { i\sqrt { \cdots } } } } } } ...
user536517
10
votes
2
answers
417
views
If $z\in\mathbb C$ with $|z|\leqslant\frac{4}{5}$, then $\sum_{n\in S}z^n\neq-\frac{20}{9}$
Let $z$ be a complex number with $|z|\le\tfrac{4}{5}$. If $S\subset\mathbb N^+$ is a finite set, then I'd like to show that
$$\sum_{n\in S}z^n\neq-\frac{20}{9}\,.$$
This problem is from an exam in ...
- 93.6k
2
votes
1
answer
399
views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\begin{align*}
\frac1n\sum_{j=0}^{n-1}\cos\left(\frac{2\pi jk}{n}\right)&=\begin{cases}
1&\text{if } k \...
- 33
2
votes
2
answers
185
views
Simplifying this (perhaps) real expression containing roots of unity
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant.
Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$.
...
- 7,789