# Tagged Questions

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### Alternating cosine series, what is the closed form?

What is the closed form for this series: $$\sum _{n=0}^{\infty } \left(\frac{\cos \left(\frac{4 \pi}{2 n+1}\right)}{2 n+1}-\frac{\cos \left(\frac{4 \pi}{2 n+2}\right)}{2 n+2}\right)$$ if any? I am ...
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### Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
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### Integrating Trigonometric Series

Let $f(x)=c_0+c_1e^{i\theta}+c_2e^{2i\theta}+...+c_ne^{ni\theta}$ where $c_k\in \mathbb R$. We need to show $$\int_{0}^{2\pi}f(e^{i\theta})\overline {f(e^{i\theta})} d\theta =2\pi\sum_{k=0}^{n} c_k$$ ...
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### How to prove that $\Sigma_{n=1}^{N-1}{\sin^2\frac{p\pi{}n}{N}} = \frac{N}{2}$

Here $p$ is a known integer constant ($p > 0$). I know that this is true for a fact (checked numerically in Matlab and it holds), but I'm just not able to prove it. I noticed a similar problem ...
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### Proof that the sum $\sum _{n=1} ^{\infty} (-1)^n \sin (nx)$ is bounded

How can I prove that there is some constant $M>0$, such that for all $N\in\mathbb{N}$ and $x\in [0,\pi]$, $$\left|\sum _{n=1} ^{N} (-1)^n \sin (nx)\right| < M\text{?}$$
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### Prove $\cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1$

$$\cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1$$ Wolframalpha shows that it is a correct identity, ...
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### Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}},$$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
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### Convergence on sum of cos [closed]

How to find the range of x on this sum to converge? $$\sum_{n=1}^∞{{\cos nx}\over{n}}.$$
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### Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since ...
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### Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
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### How to find the sum of a geometric progression involving cos using complex numbers?

Use $2\cos{n\theta} = z^n + z^{-n}$ to express $\cos\theta + \cos3\theta + \cos5\theta + ... + \cos(2n-1)\theta$ as a geometric progression in terms of $z$. Hence find the sum of this progression ...
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### Extensions of Ramanujan's Ramanujan Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is as stated here . Then there is a line . Equating coefficients of $\theta^0$, $\theta^4$, and $\theta^8$ gives some amazing identities for the hyperbolic secant. ...
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### Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
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### Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
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### Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
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### Looking for a source of an infinite trigonometric summation and other such examples.

Question: If $x \neq 0$, then prove that $\displaystyle \sum_{n=1}^{\infty}\dfrac1{2^n} \tan\left(\dfrac{x}{2^n}\right) = \dfrac1{x} - \cot x.$ My answer: I proved this result by using the ...
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### Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
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### Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$

Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
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By looking through an book, I found this interesting series To prove that: $$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ... 1answer 107 views ### finding \lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right) I want to find$$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$I already know that I need to split the expression \frac{2r}{1-r^2+r^4} of the form ... 0answers 173 views ### On the series  \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) . Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it?$$ \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
If you have $\sum_{n = 0}^\infty(4/5)^n$ and you are asked to represent it as a geometric series you would: $\sum_{n = 0}^\infty(4/5)(4/5)^{n-1}$ //factor out your constant therefore $a = 4/5$, ...
### Calculate: $\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$
Calculate the following sum for integers $n\ge2$: $$\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$$ I'm trying to obtain a closed form if that is possible.