# Parseval's Theorem

1. Apr 23, 2014

### AppleFritters

1. The problem statement, all variables and given/known data
I'm given the following function
$$f(x) = \begin{cases} x &-2<x<2\\ f(x+4) &\mbox{otherwise} \end{cases}$$

And I'm asked to find the Fourier sine series. Then I'm supposed to use Parseval's theorem to obtain a certain sum.

2. Relevant equations
Since I have a sine Fourier series, Parseval's Theorem for this says
$$\frac{1}{b-a} \int_a^b |f(x)|^2 dx = \sum_{n=1}^\infty b_n^2$$

3. The attempt at a solution
So I worked through and got the Fourier sine series for this function which is
$$f(x) = \frac{4}{\pi} \left[ \sin \left( \frac{\pi x}{2} \right) - \frac{1}{2} \sin \left( \frac{2 \pi x}{2} \right) + \frac{1}{3} \sin \left( \frac{3 \pi x}{2} \right) - \frac{1}{4} \sin \left( \frac{4 \pi x}{2} \right) + \ldots \right]$$

Now I apply Parseval's Theorem:
$$\frac{1}{b-a} \int_a^b |f(x)|^2 dx = \sum_{n=1}^\infty b_n^2 \\ \frac{1}{2-(-2)} \int_{-2}^2 x^2 dx = \sum_{n=1}^\infty b_n^2 \\$$

On the left hand side:
$$\frac{1}{4} \int _{-2}^{2} x^2 dx = \frac{16}{12}$$

On the right hand side:
$$\left(\frac{4}{\pi} \right)^2 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{16}{\pi ^2} \sum_{n=1}^\infty \frac{1}{n^2}$$

Now equating the left and right hand sides:
$$\frac{16}{12} = \frac{16}{\pi ^2} \sum_{n=1}^\infty \frac{1}{n^2}\\$$
$$\frac{\pi ^2}{12} \sum_{n=1}^\infty \frac{1}{n^2}$$

The answer I'm supposed to be getting is
$$\frac{\pi ^2}{6} \sum_{n=1}^\infty \frac{1}{n^2}$$

So I'm off by a factor of a half somewhere but I can't figure it out. Some help would be appreciated. Thank you.

2. Apr 24, 2014

### jbunniii

I don't think you have stated Parseval's theorem correctly for the sine series.

Parseval's theorem for the complex Fourier series is as follows:
$$\sum_{n=-\infty}^{\infty}c_n e^{2\pi i n x / P}$$
in which case
$$\sum_{n=-\infty}^{\infty}|c_n|^2 = \frac{1}{b-a}\int_{a}^{b} |f(x)|^2 dx$$
If you have a real Fourier series:
$$\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(2\pi n x / P) + \sum_{n=1}^{\infty} b_n \sin(2\pi n x / P)$$
then how does $c_n$ relate to $a_n$ and $b_n$?

Last edited: Apr 24, 2014