# Tagged Questions

For questions about or related to trigonometric series.

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### The number of roots of a truncated trigonometric series

If $f$ is continuous on $(0,\pi)$ and $\int \limits _0 ^\pi f(x)\cos kx \, dx = \int \limits _0 ^\pi f(x)\sin kx \, dx=0$ for each $0 \le k \le n$, prove that $f$ has $2n$ different zeros in the ...
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### Sum into closed form

I'm working with spectral approximations and I ran into this problem. Hope someone knows how to solve it! $(D_N)_{lj} = \frac{1}{N} \sum_{k=-N/2}^{N/2-1} i k e^{2 i k (l-j) \pi /N}$ ...
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### Sum up trigonometric series [duplicate]

$$\cos \frac{2π}{2013} +\cos \frac{4π}{2013} +\cdots+\cos \frac{2010π}{2013} + \cos \frac{2012π}{2013}$$ How to sum it up? *Calculator is not allowed.
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### What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
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### How to prove $\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$ for any n>1?

I can show for any given value of n that the equation $$\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$$ is true and I can see that geometrically it is true. However, I can not seem to prove it out ...
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### Convergence of a sine series

Using Mathematica, we claim that the following series is convergent: $$\sum_{n=1}^{\infty}\frac{\sin(n^2 t)}{n}$$ Any idea how we prove this?
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### On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
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### prove that $\sum_{k=1}^{n}{\sin(kA)}$ [duplicate]

prove that $$\sum_{k=1}^{n}{\sin(kA)} = {{\cos({A\over2})-\cos(nA+{A\over 2})}\over 2\sin({A\over 2})}$$using Telescoping series. How do i go about doing this?
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### Infinite sum of cosine function [closed]

What does the following expression equal to $$\sum\limits_{n=1}^\infty \cos(n\cdot\theta)=\text{?}$$
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### Find the limit of $\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

Can anyone give a hint on how to see if the following has a limit? $f$ stands for frequency. $$\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$$ I've tried a few ...
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### Solving a question on trigonometric series

I have stated the sum of the sum of the series (by geometric series) which is $$S_n= \frac{z(1-z^n)}{1-z}$$ I am trying to prove the second part of the question. However, I am unable to reach to ...
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### 1D diffusion equation with Robin boundary conditions

Solving $u_t = \alpha^2 u_{xx}$ with boundary and initial conditions $u(0,t)=0$, $u_x(1,t)+h u(1,t)=0$, $u(x,0)=x$. (Following the book by Farlow, "PDEs for scientists and engineers", page 54) ...
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### Can I use a change of variable to simplify this series of Sines?

I have a function that looks something like: A sin(x)+B sin(2x)+C sin(3x)+D sin(4x)+...+M sin(Nx) Is there some change of variable I can use to turn this into a polynomial? Or more generally - are ...
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### simplify the summation of fraction of two sinusoidal functions

How can I simplify the expression bellow $$\sum_{r=0}^{N-1}\frac{\sin^2(\pi\epsilon)}{\sin^2\bigg(\dfrac{\pi(r-n+\epsilon)}{N}\bigg)}$$ where $n$ and $N$ are integers? Does this summation equal ...
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### Dirichlet kernel [duplicate]

Can anybody help me please in showing that $$\dfrac{1}{2} + \sum_{k = 1}^{N} \cos(kx) = \dfrac{\sin\left(N + \dfrac{1}{2}\right)}{2\sin\dfrac{x}{2}}$$ please?
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### $\sin x$ as a sum involving fractional parts

Does there exist a formula giving a sense to the formal equation $$\sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}\left\{\frac{nx}{2\pi}\right\},$$ where $\mu$ is the Möbius function, $\{\cdot\}$ ...
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### Computing the limit of a series involving trigonometric identities

Let $x_n\in(n\pi,(n+1)\pi)$ with $\tan(x_n)=x_n$. Can $$s:=\sum_{n=1}^\infty \frac{1}{x_n^2}$$ be determined explicitly? My ideas so far: First, I wrote $x_n$ as $$x_n= \pi (n+z_n)$$ with ...
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### Sine/cosine series

$$\frac{\sin²(1°) + \sin²(2°) + \sin²(3°) + .. + \sin²(90°)}{\cos²(1°) + \cos²(2°) + \cos²(3°) + .. + \cos²(90°)} = ?$$ I tried to use multiple identities but I couldn't simplify the expression. ...
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### Values of the sums $\sum\limits_{k=1}^{n}\cos^4(πk/(2n+1))$

I have been given a question which asks you to prove that $$\sum_{k=1}^{n}\cos^4\left(\frac{πk}{2n+1}\right)=\frac{6n-5}{16}$$ The main problem I have with solving this is that since the summands ...
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### When does $-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$ where $z$ is a complex variable?

Let $z$ be a complex variable. Is there someone who can show me when does :$$-\frac{\pi z}{2}\cot(\pi z)+\frac{1}{2}=0$$ Note: I have tried using trigonometric formulas but it didn't work. Maybe I ...
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### Is there any trigonometric function that cannot be written as an infinite series?

Let $p_n(x)=x^n$ for $x\in \Bbb{R}$ and let P=span$\{p_0,p_1,p_2,p_3\dots\}$. Then- P is the vector space of all real valued continuous functions on R P is a subspace of all real ...
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### Do they have a mistake in this heat equation?

I need to know if there is a mistake in these notes: In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a ...