Fourier Series - Wave Eqn

I just have a quick question:

I'm evaluating a Fourier coefficient for the following series:

Where and

I'm going to show my work:

I said that

This is what I did to simplify what was in the brackets:

Multiplying by , I get:

Now this is where I'm somewhat in the dark. Since I'm plugging in values of n, where , into , I notice when n is odd, I get either 1 or -1, and when n is even, I get zero. Thus, I rewrote my answer as:

I noticed that could be written as

So then I saw that

Thus, my series solution would be

Is this correct??

However, my question is this: when I rewrote as , do I need to replace the other n's in my equation with 2n-1? meaning, would my answer for be ????

I'd appreciate any help.