# Using a Fourier Cosine Series to evaluate a sum

1. Mar 3, 2014

### richyw

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

a) Show that the Fourier Cosine Series of $f(x)=x,\quad 0\leq x<L$ is
$$x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right]$$

b) use the above series to evaluate the sum$$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...$$

2. Relevant equations

Fourier Cosine Series General form

3. The attempt at a solution

So I have done part a, but I am lost on how to do part b. I don't understand how to get the even terms? Perhaps I need to differentiate the series, which would pull an n out, making the n's even.

To differentiate term by term a cosine series I need f'(x) to be piecewise continuous. Which I think it is.

Last edited: Mar 3, 2014
2. Mar 3, 2014

### micromass

Staff Emeritus
Fixed LaTeX:

3. Mar 3, 2014

### richyw

thanks. I did too. I had to quickly reinstall LaTeXiT from macports

4. Mar 3, 2014

### richyw

but my fourier series only has the odd-n terms?

5. Mar 3, 2014

### micromass

Staff Emeritus
I'll assume the Fourier expansion is correct.

What you need to do is to choose $x$ (and perhaps $L$) wisely in order to extract an interesting series.

Now, also note the following:

Take

$$S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + ...$$

then

$$S = \left(1 + \frac{1}{3^2} + \frac{1}{5^2} + .... \right) + \left(\frac{1}{2^2} + \frac{1}{4^2} + ...\right)$$

But

$$\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + ...= \frac{1}{4}\left(1 + \frac{1}{2^2}+ \frac{1}{3^2} + ...\right)= \frac{S}{4}$$

So differentiating is unnecessary.

6. Mar 3, 2014

### richyw

ok my initial guess was wrong obviously. becuase it would not make the n's even, it would make them not squared...

7. Mar 3, 2014

### vanhees71

First think about, what you get, setting $x=L$. Then you can think further about how to get the given series from this!

8. Mar 3, 2014

### richyw

ok. I don't really have time to show my work, but I ended up with π^2/6. Is this the correct answer?

the trick I didn't get was that part with the s/4

9. Mar 3, 2014

### micromass

Staff Emeritus
Actually, this is the Fourier series of the even function $f(x) = |x|, -L \leq x \leq L$, extended to the whole real line as a periodic function of period $2L$. Basically, the person posing the question needs to specify $f(x)$ outside the desired interval $[0,L]$. If that is not done correctly the Fourier series won't be pure 'cosine' and/or might not be continuous at $\pm \, L$, leading to a mis-match between f and the sum of the series at those points.