# Help regarding a question Parseval's Theorem

1. Oct 29, 2008

### TFM

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

The Fourier series for f(x) = x2 over the interval (−1/2, 1/2) is:

$$f(x) = \frac{1}{12}-\frac{1}{\pi^2} (cos 2\pi x - \frac{1}{2^2}cos4\pi x + \frac{1}{3^2}cos6\pi x) ...$$

Using Parseval's Theorem, show that

$$\sum _{n = 1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$$

2. Relevant equations

Fourier's Series:

$$f(x) = \frac{1}{2}a_0 + \sum _{n = 1}^\infty a_n cos\frac{2\pi nx}{l} \sum _{n = 1}^\infty b_n sin \frac{2\pi nx}{l}$$

$$a_0 = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) dx$$

$$a_n = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) cos \frac{2\pi nx}{l} dx$$

$$b_n = \frac{1}{l} \int^{\frac{l}{2}}_{\frac{-l}{2}}f(x) sin \frac{2\pi nx}{l} dx$$

Parseval's Theorem:

$$\frac{1}{2\pi} \int^{\pi}_{-\pi}f(x)^2 dx = \frac{1}{4}a_0^2 + \frac{1}{2} \sum _{n = 1}^\infty a_n^2 + \frac{1}{2}\sum _{n = 1}^\infty b_n^2$$

3. The attempt at a solution

See I'm not quite sure where to go from here. It says that it is a Fourier series, but it doesn't seem to fit with the definition of Fourier Series I have quoted below?

Any assitance would be greatly appreciated,

TFM

2. Oct 29, 2008

### Office_Shredder

Staff Emeritus
You multiplied the cosine and sine portions of the Fourier series together, they're supposed to be added. At any rate

For your f(x), a0=1/6 and then you can distribute the $$\frac{-1}{\pi^2}$$ to calculate the other ai's

And then the sine component is just all zeroes (so all your bi's will be 0)

3. Oct 29, 2008

### TFM

So firstly,

$$f(x) = \frac{1}{2}a_0 + \sum _{n = 1}^\infty a_n cos\frac{2\pi nx}{l} + \sum _{n = 1}^\infty b_n sin \frac{2\pi nx}{l}$$

And

$$a_0 = \frac{1}{6}$$

As there are no sins, this implies that b_n = 0 (since b_n is the sin component.)
Multiplying out the brackets:

$$f(x) = \frac{1}{12} - (\frac{1}{\pi^2}cos 2\pi x - \frac{1}{\pi^2}\frac{1}{2^2}cos4\pi x + \frac{1}{\pi^2}\frac{1}{3^2}cos6\pi x ...)$$

$$f(x) = \frac{1}{12} - (\frac{1}{\pi^2}cos 2\pi x - \frac{1}{4\pi^2}cos4\pi x + \frac{1}{9\pi^2}cos6\pi x ...)$$

So the a_n seems to be an increasing value:

$$a_n = frac{1}{\pi^2}, frac{1}{4\pi^2}, frac{1}{9\pi^2}...$$

Does this seem okay so far?

???

TFM