# Find SUM for fourier series

1. Oct 1, 2012

### ensten

1. The problem statement, all variables and given/known data
$h(x)=\left\{\begin{matrix} 9+2x , 0<x<\pi\\ -9+2x , -pi<x<0 \end{matrix}\right. \\ Find \ the \ sum \ of \ the \ fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi \\ The \ fourier \ series \ is: \\ h(x)=9+\pi + \sum_{n=1}^{inf} \frac{18-2(9+2\pi)(-1)^2}{n\pi}sin(nx)$
Also period is 2pi.

3. The attempt at a solution
I have already calculated the fourier series. I however don't know how I can find the sum for the x's. I have searched the internet and looked in my book but I can't find any examples that help me. I'm stuck and don't know how to go from here.

2. Oct 1, 2012

### jbunniii

Can you simplify $\sin(n \frac{3\pi}{2})$ and $\sin(n \pi)$?

3. Oct 1, 2012

### ensten

$\sin(n \frac{3\pi}{2})$ changes for different n's. Example: -1, 0 and 1. So I don't know how to simplify that. However $\sin(n \pi)$ will always be 0 for all n.

4. Oct 1, 2012

### jbunniii

$\sin(n \frac{3\pi}{2})$ is zero for even $n$. Try a change of variables so that you will only sum over the odd values of $n$.

5. Oct 1, 2012

### ensten

I don't quite understand how to do that. I tried the sum function at wolframalpha and I get that it doesnt converge.

6. Oct 1, 2012

### LCKurtz

You don't work this kind of problem by actually summing the series. You use the convergence theorem. Doesn't your text have a theorem something to the effect that for a periodic function f(x) satisfying the Dirichlet conditions the series converges to f(x) at points where f is continuous and to $\frac {f(x^+) + f(x^-)} 2$ at points $x$ where there is a jump discontinuity? So you can answer the question by examining the periodic extension of $f(x)$. In fact, you don't even need to calculate the Fourier Series to answer the question.