Computing the Fourier Series for f(x)=x^2

[kieranl]

Homework Statement

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

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))

 

[Cyosis]

Yep that is correct.

 

[kieranl]

cheers

 

[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?

 

[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.

 

[kieranl]

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

 

[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.

 

[jbunniii]

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.

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

 

[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.