# Fourier series of a function defined by parts

• Oct 30th 2009, 01:24 AM
javicg
Fourier series of a function defined by parts
I'm having trouble with the following question. Any help is appreciated.

Calculate the Fourier series for

$\displaystyle f(x) = \left\{\begin{array}{rl} x^2, & 0 \le x \le \pi \\ -x^2, & -\pi \le x \le 0, \end{array}\right.$

and $\displaystyle f (x+2\pi) = f(x), \forall x \in \mathbb{R}$. Given that $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^4} = \frac{n^4}{48},$ find the value of $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^6} = \sum_{n=1}^{\infty} \frac1{(2n+1)^6}$
• Oct 30th 2009, 04:28 AM
CaptainBlack
Quote:

Originally Posted by javicg
I'm having trouble with the following question. Any help is appreciated.

Calculate the Fourier series for

$\displaystyle f(x) = \left\{\begin{array}{rl} x^2, & 0 \le x \le \pi \\ -x^2, & -\pi \le x \le 0, \end{array}\right.$

and $\displaystyle f (x+2\pi) = f(x), \forall x \in \mathbb{R}$. Given that $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^4} = \frac{n^4}{48},$ find the value of $\displaystyle \sum_{n=1}^{\infty} \frac{1 - (-1)^n}{n^6} = \sum_{n=1}^{\infty} \frac1{(2n+1)^6}$

What exactly is/are the problem/s?

You know the definition of the Fourier series, and you have the definition of the function, what have you done with this knowledge?

Also the last sentence does not appear to be connected to the rest of the question.

CB