All Questions
41
questions
25
votes
5
answers
2k
views
Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
In old popular science magazine for school students I've seen problem
Prove that $\quad $
$\dfrac{1}{\cos^2 20^\circ} +
\dfrac{1}{\cos^2 40^\circ} +
\dfrac{1}{\cos^2 60^\circ} +
\dfrac{1}{\cos^...
- 17.4k
21
votes
7
answers
18k
views
Finite Sum $\sum\limits_{k=0}^{n}\cos(kx)$
I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$
I have some progress made, but I am stuck and could use some help.
What I did:
It ...
- 23.2k
18
votes
2
answers
497
views
Prove $\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \frac n4$
How do you prove that $\displaystyle\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \dfrac{n}{4}$, where $x\in\mathbb{R}$?
I tried mathematical induction, but it doesn't work. I also ...
- 405
10
votes
3
answers
12k
views
Proof of $\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\frac12n\theta}{\sin\frac12\theta}\cos\frac12(n+1)\theta$
State the sum of the series $z+z^2+z^3+\cdots+z^n$, for $z\neq1$.
By letting $z=\cos\theta+i\sin\theta$, show that
$$\cos \theta+\cos 2\theta+\cos 3\theta+\cdots+\cos n\theta=\frac{\sin\...
- 1,927
7
votes
3
answers
3k
views
Cotangent summation (proof)
How to sum up this thing, i tried it with complex number getting nowhere so please help me with this,$$\sum_{k=0}^{n-1}\cot\left(x+\frac{k\pi}{n}\right)=n\cot(nx)$$
- 164
7
votes
3
answers
157
views
History of the general formula for linearising $\cos^n(x)$
I was wondering where the formula:
$$\cos^n(x)=\sum_{k=0}^n\frac{n!}{k!(n-k)!}\cos(x(2k-n))$$
Was first originally published. I accidentally derived it a few days ago and was wondering where it was ...
- 584
7
votes
2
answers
305
views
Bounding a sum involving a $\Re((z\zeta)^N)$ term
This is a follow up to this question. Any help would be very much appreciated.
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$.
Let $\...
- 7,789
6
votes
2
answers
2k
views
Prove that $\cos \frac{2\pi}{2n+1}+\cos \frac{4\pi}{2n+1}+\cos \frac{6\pi}{2n+1}+...+\cos \frac{2n\pi}{2n+1}=\frac{-1}{2}$
Prove that $$\cos \frac{2\pi}{2n+1}+\cos \frac{4\pi}{2n+1}+\cos \frac{6\pi}{2n+1}+...+\cos \frac{2n\pi}{2n+1}=\frac{-1}{2}$$
My attempt,
Let an equation $x^{2n+1}-1=0$, which has roots $$\cos \frac{...
- 7,798
4
votes
1
answer
263
views
How to calculate $\sum_{k=0}^n a^k\sin(kx)$?
I tried to evaluate
$$
\sum_{k=0}^n a^k\sin(kx)
$$
using complex numbers but it didn't work... Any hint?
$a$ and $x$ are real numbers.
- 145
4
votes
0
answers
128
views
How to solve $\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}$?
I need to solve this sum:
$$\sum_{n=-\infty}^\infty\frac{y^2}{[(x-n\pi)^2+y^2]^{3/2}}.$$
Do you have any ideas for how I could do this?
I know that this sum:
$$\sum_{n=-\infty}^\infty\frac{y}{(x-n\pi)^...
- 1,027
3
votes
4
answers
174
views
Evaluate $\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}$, where $q^7=1$ and $q\neq 1$.
Let $q$ be a complex number such that $q^7=1$ and $q\neq 1$. Evaluate $$\frac{q}{1+q^2}+\frac{q^2}{1+q^4}+\frac{q^3}{1+q^6}.$$
The given answer is $\frac{3}{2}$ or $-2$. But my answer is $\pm 2$.
...
- 373
3
votes
1
answer
359
views
Finding the sum of $\cos{x}+2\cos{2x}+...+n\cos{nx}$ [duplicate]
I'm struggling to find the sum of $S_n=\cos{x}+2\cos{2x}+3\cos{3x}...+n\cos{nx}$
I know that for $z=e^{ix}$, $2\cos{nx}=z^n+\frac{1}{z^n}$.
So I've tried $2S_n=(z+2z^2+3z^3+...+nz^n)+(\frac{1}{z}+\...
- 1,381
3
votes
1
answer
1k
views
Understanding a step in applying deMoivre's Theorem to $\sum_{k=0}^n \cos(k\theta)$
I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far:
\begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} \...
- 231
2
votes
1
answer
86
views
Find the sum of infinite series $\cos{\frac{\pi}{3}}+\frac{\cos{\frac{2\pi}{3}}}{2}+..$
Find the sum of infinite series $$\cos{\dfrac{\pi}{3}}+\dfrac{\cos{\dfrac{2\pi}{3}}}{2}+\dfrac{\cos{\dfrac{3\pi}{3}}}{3}+\dfrac{\cos{\dfrac{4\pi}{3}}}{4}+\dots$$
I tried to convert it to $\mathrm{cis}$...
- 1,157
2
votes
2
answers
97
views
How do I prove that $\sum_{i=1}^m \cos^2\left(\frac{2\pi i}{m}\right) = \sum_{i=1}^m \sin^2\left(\frac{2\pi i}{m}\right) = \frac{m}{2}$? [duplicate]
I haven't had any good ideas nor found any helpful identities so far, so I'd appreciate some help.
Also, here $m > 2$.
Update: Thanks to the hints and to this previous post I managed to get to ...
- 6,833