something is wrong with what I am doing. I'm either:
- Using the wrong definition of something(but from Wikipedia, I don't think I am).
- Did some mistake in my calculations (also don't think so)
So I was trying to do this: Evaluate $\sum^\infty_{n=1} \frac{1}{n^4} $using Parseval's theorem (Fourier series)
What I started doing:
$$a_0= \frac{1}{\pi} \int_{-\pi}^{\pi} x^2dx= \frac{2}{\pi} \int_{0}^{\pi} x^2\,dx =\frac{2 \pi^2}{3}$$ here it is already different from that topic. But from wikipedia and from my classes this is what I am supposed to be calculating. I suppose I'm doing something wrong in the definition of the parseval identity. Using my definition of $a_0$ am I suppose to sum $a_0^2 + \sum_{n=1}^{\infty} a_n^2$, $\frac{a_0^2}{2}+ \sum_{n=1}^{\infty} a_n^2$ or $\frac{a_0^2}{4}+\sum_{n=1}^{\infty} a_n^2$ ?
For clarification I define: $$a_n=\frac{1}{\pi} \int_{-\pi}^{\pi}f(x)\cos(nx)dx$$ whenever $f$ is defined in $[-\pi,\pi]$
My fourier series is then written as: $$f=\frac{a_0}{2} + \sum_{n=1}^{\infty}(a_n\cos(nx)+b_n\sin(nx))$$
I know that in this specific case $b_n$ is zero.
So what I am doing wrong?
EDIT: just for the sake of completeness my $a_n$ was also off by a factor of 2, it was $a_n= \frac{4 (-1)^n}{n^2}$