Fourier Series/Summation Question

[erok81]

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]

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?

 

[erok81]

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]

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.