# Fourier Series/Summation Question

I'm not going to use the standard question as I've already solved it, this is more of a general question that doesn't fit into the three question format.

My question arises almost at the very end of the Fourier Series. It has been a looooong time since I've done summations so I am not sure if this is how they work.

Anyway....I have this:

$$\frac{p^2}{3}+\sum \frac{4p^2 cos(\pi n)}{\pi^2 n^2}$$

The 4p^2/Pi^2 can come out to the front and the sign always seems to change when it does. So the final answer is this:

$$\frac{p^2}{3}-\frac{4p^2}{\pi^2}\sum \frac{cos(\pi n)}{\n^2}$$

So, does the sign flip when you pull it out of the summation? Maybe one subtracts that from the sum and that is why the sign changes?

On a completely separate Fourier Series note. It seems whenever I have sin(pi*n) I can set that equal to zero for the series (at least I can get my answers to match that way). Is that a correct method? I know for any value of n it would be zero, but I want to make sure that is what is happening.

Hopefully that all makes sense. Thanks for the help.

Mark44
Mentor
Starting with this:
$$\frac{p^2}{3}+\sum \frac{4p^2 cos(\pi n)}{\pi^2 n^2}$$

and pulling out the constant from the summation, you should get this:
$$\frac{p^2}{3} + \frac{4p^2}{\pi^2}\sum \frac{cos(\pi n)}{n^2}$$

I'm assuming the summation is from n = 0 to infinity. There is no reason for the sign to change when you bring 4p2/(pi)2 out. Why would you think this?

Can you show an example of where you think this happened?

Starting with this:
$$\frac{p^2}{3}+\sum \frac{4p^2 cos(\pi n)}{\pi^2 n^2}$$

and pulling out the constant from the summation, you should get this:
$$\frac{p^2}{3} + \frac{4p^2}{\pi^2}\sum \frac{cos(\pi n)}{n^2}$$

I'm assuming the summation is from n = 0 to infinity. There is no reason for the sign to change when you bring 4p2/(pi)2 out. Why would you think this?

Can you show an example of where you think this happened?

The summation was from n=1 to infinity. I couldn't figure out how to add those with latex.

I agree with you, I didn't make sense but it seemed following the solution manual it's the only way I could get the correct answer.

Let me see if I can dig up the example that it happened with and I'll post it.

Mark44
Mentor
Here are your two summation expressions with the limits. Click either one to see how the limits should look.
$$\frac{p^2}{3}+\sum_{n = 1}^{\infty} \frac{4p^2 cos(\pi n)}{\pi^2 n^2}$$
$$= \frac{p^2}{3}+\frac{4p^2}{\pi^2}\sum_{n = 1}^{\infty} \frac{ cos(\pi n)}{ n^2}$$

The property used here is
$$\sum_{n = 1}^{\infty}k f(n) = k\sum_{n = 1}^{\infty}f(n)$$

Here k is a constant, so doesn't involve n.