# 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.