All Questions
Tagged with summation trigonometry
87
questions
4
votes
1
answer
108
views
Clean proof for trigonometry identity? I know what the answer is, but I feel like there should be like a $1$-$2$ liner to compute this
Fix $j,k$ with $0 \leq j,k \leq N$. If $j+k$ is even, (i.e. if $j,k$ have same parity), then
$$
\sum_{n=1}^{N-1} \cos\left(\frac{j\pi}Nn\right)\sin\left(\frac{k \pi}Nn\right) = 0
$$
and
$$
\sum_{n=0}^...
- 175
3
votes
1
answer
123
views
How should I prove that: $\sum_{i=1} ^{n}(\sin(\frac{i\pi}{n}))^2=\frac{n}{2}$
$$\sum_{i=1} ^{n}\Big(\sin\big(\frac{i\pi}{n}\big)\Big)^2=\frac{n}{2}$$
An interesting conclusion and checked for validity...holds for $n\geq 2$, but yet do not know how to prove it. Are there any ...
- 345
3
votes
1
answer
383
views
How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$ [duplicate]
How prove that sum
$$\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$$
- 6,320
2
votes
1
answer
189
views
Evaluating $\sum_{n=1}^\infty \frac{\sin(nx)}{n}$ without integrating $\sum_{n=1}^\infty e^{nx}$
I am looking for alternative solutions for finding this sum
$$\sum_{n=1}^\infty \frac{\sin(nx)}{n} $$
My solution proceeds by integrating $$\sum_{n=1}^\infty e^{nx}=\frac{e^{ix}}{1-e^{ix}}$$
With ...
- 313
2
votes
3
answers
256
views
Evaluate $\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$
I would like to know how to evaluate $$\sum\limits_{r=1}^\infty(-1)^{r+1}\frac{\cos(2r-1)x}{2r-1}$$
There are a couple of issues I have with this. Firstly, depending on the value of $x$, it seems, at ...
- 6,560
2
votes
1
answer
224
views
Do there exist $a_k$ and $b_k$ so the equation $\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$ has no roots?
Do there exist real numbers $a_1, a_2, ..., a_n$ and $b_1, b_2, ..., b_n$ such that the equation
$$\sum\limits_{k=1}^{n} (a_k \sin(kx) + b_k \cos(kx)) = 0$$
has no solutions?
- 3,269
2
votes
2
answers
185
views
Simplifying this (perhaps) real expression containing roots of unity
Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant.
Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$.
...
- 7,789
2
votes
1
answer
399
views
A finite sum of trigonometric functions
By taking real and imaginary parts in a suitable exponential equation, prove that
$$\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
1
answer
276
views
$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = ?$
Prove that (not use induction)
$\displaystyle\sum_{k=0}^n \frac{\cos(k x)}{\cos^kx} = \frac{1+(-1)^n}{2\cos^nx} + \dfrac{2\sin\big(\lfloor\frac{n+1}{2}\rfloor x\big) \cos\big(\lfloor\frac{n+2}{2}\...
- 1,520
1
vote
1
answer
183
views
Telescoping series: $\sum\limits_{n=1}^{∞}[\tan^{-1}(2n+1)-\tan^{-1}(2n-1)]$
In this question that was asked today the OP wrote that
$$\begin{align}\sum\limits_{n=1}^{∞}[\arctan(2n+1)-\arctan(2n-1)]&=\arctan\infty-\arctan 1\\
&=\frac{\pi}{2}-\frac{\pi}{4}\\&=\frac{\...
- 6,560
1
vote
4
answers
396
views
Proving a function is continuous and periodic
Suppose we are given a function
$$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$
and
$$g\left ( k\pi \right )=\...
- 485
1
vote
2
answers
18k
views
Induction proof of the identity $\cos x+\cos(2x)+\cdots+\cos (nx) = \frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}$ [duplicate]
Prove that:$$\cos x+\cos(2x)+\cdots+\cos (nx)=\frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}.\ (1)$$
My attempt:$$\sin\left(\frac{x}{2}\right)\sum_{k=1}^{n}\cos{(kx)}$$$$=\sum_{k=1}^...
- 123
1
vote
3
answers
233
views
Calculate $\sum_{n=1}^{\infty} \arctan\bigl(\frac{2\sqrt2}{n^2+1}\bigr) $
$$ \lim_{n \to\infty} \sum_{k=1}^{n} \arctan\frac{2\sqrt2}{k^2+1}= \lim_{n \to\infty} \sum_{k=1}^{n} \arctan\frac{(\sqrt{k^2+2}+\sqrt2)-\sqrt{k^2+2}-\sqrt2)}{(\sqrt{k^2+2}+\sqrt2)(\sqrt{k^2+2}-\sqrt2)+...
- 1,219
1
vote
1
answer
120
views
Find the value of sum $\forall\:\:\alpha,\beta\in\mathbb{R}$
Evaluate the sum $\forall\:\:\alpha,\beta\in\mathbb{R}$ $$S=\sum_{n=1}^{\infty}\frac{\alpha^{n+1}-1}{n(n+1)}\sin\left(\frac{n\pi}{\beta}\right)$$
I rewrote this as $$S=\sum_{n=1}^{\infty}\left(\left(\...
1
vote
1
answer
5k
views
Summation of $\cos (2n-1) \theta$
By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that
$$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$
where $\sin\theta\neq0$
I ...
- 1,927