# Fourier series convergence test

1. Sep 10, 2011

### JamesGoh

1. The problem statement, all variables and given/known data

A function f(x) is given as follows

f(x) = 0, , -pi <= x <= pi/2
f(x) = x -pi/2 , pi/2 < x <= pi

determine if it's fourier series (given below)

$F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx) - (1/(2n^{2}))( n\pi cos(n\pi) + 2sin(n\pi/2) )sin(nx) ]$

for n= 1 to infinity

converges to it for the case of x = pi/2

2. Relevant equations

F(x) = 0.5*[f(x+) + f(x-) ]

3. The attempt at a solution

see pdf attachment

#### Attached Files:

• ###### q37a.pdf
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2. Sep 12, 2011

### vela

Staff Emeritus
Your work is kind of hard to follow, but it looks like you incorrectly got rid of the cosine terms. You have

$$F(x)=\frac{\pi}{16} + \frac{1}{\pi}\sum_{n=1}^\infty \frac{\cos n\pi -\cos \frac{n\pi}{2}}{n^2} \cos nx - \frac{1}{\pi}\sum_{n=1}^\infty \frac{n\pi \cos n\pi + 2\sin \frac{n\pi}{2}}{2n^2}\sin nx$$

You can simplify it a bit by noting that $\cos n\pi = (-1)^n$. Next, when $x=\pi/2$, the values of cos nx and sin nx depend on whether n is odd or even. Use those facts to get the series for $F(\pi/2)$.

Last edited: Sep 12, 2011
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