# Fourier series

1. May 20, 2009

### kieranl

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

Compute the Fourier series for the given function f on the specified interval
f(x) = x^2 on the interval − 1 < x < 1

3. The attempt at a solution

Just wondering if anyone can verify my answer?

f(x)=1/3+$$\sum$$(4/(n^2*pi^2)*(-1)^n*cos(n*pi*x))
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. May 20, 2009

### Cyosis

Yep that is correct.

3. May 20, 2009

### kieranl

cheers

4. May 20, 2009

### kieranl

the next part of the question says to determine if the function to which the fourier series for f(x) converges?? does this make sense to anyone?

5. May 20, 2009

### Cyosis

Is that exactly how the question is asked in your text? The "if" confuses me. I guess what they are asking you is if the Fourier series you just derived converges. Which is pretty easy to show.

6. May 20, 2009

### kieranl

how do you show that it converges?? just pick example numbers??

7. May 20, 2009

### Cyosis

That would mean you would have to pick an infinite amount of numbers though. To show that it converges compare it to a series you know to be convergent. For example, $\sum_{n=1}^\infty \frac{1}{n^2}$. You could also try the root test and or ratio test.

8. May 20, 2009

### jbunniii

You should even be able to show uniform convergence if you use the right test.

9. Jun 10, 2009

### txy

there are theorems that state that when the function satisfies certain conditions, the Fourier series of the function converges to some expression. if you have learned these theorems, then it's quite easy to show that the Fourier series converges to the function itself.