All Questions
4
questions
3
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4
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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 $\pm 2$.
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1
vote
3
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106
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If $n>3$ prove that $\sum_{k=0}^{n-1} (k-n)\cos\frac{2k\pi}{n}=\frac{n}{2}$.
Do you have any ideas on this IIT exercise?
If $n>3$ is an integer, prove that
$$\sum_{k=0}^{n-1} (k-n)\cos(2kπ /n) = n/2$$
In my attempt, I have considered
$$z=cis(2kπ/n), k=[1, 2,..., n-1]...
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7
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2
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305
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Bounding a sum involving a $\Re((z\zeta)^N)$ term
This is a follow up to this question. Any help would be very much appreciated.
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$.
Let $\...
- 7,789
2
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185
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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 $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$.
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