Using a Fourier Cosine Series to evaluate a sum

[richyw]

Homework Statement

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

Homework Equations

Fourier Cosine Series General form

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.

 

[micromass]

Fixed LaTeX:

Homework Statement

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

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

Homework Equations

Fourier Cosine Series General form

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.

 

[richyw]

Fixed LaTeX:

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

 

[richyw]

Anyway, what you need to do is choose $x$ (and perhaps $L$) suitably such that you can extract the series $\sum \frac{1}{n^2}$ out of there.

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

 

[micromass]

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.

 

[richyw]

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

 

[vanhees71]

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

 

[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

 

[micromass]

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

You have the correct answer!

http://en.wikipedia.org/wiki/Basel_problem

 

[Ray Vickson]

Fixed LaTeX:

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.