With 
, the Fourier series expansion of 
is
where the coefficients are obtained by computing
You should end up with
(both due to the fact that is odd)
Now the problem is that this expansion does not match the given one. As a matter of fact, since is odd, there is no cosine series. So I'm starting to think this question is missing some initial details.
One possibility is that you're actually supposed to use the even extension of, which is to say we're actually considering the function
and enforcing a period of
. Now, you should find that
The value of the sum can then be verified by choosing
, which gives
as required.
where the coefficients are obtained by computing
You should end up with
(both due to the fact that
Now the problem is that this expansion does not match the given one. As a matter of fact, since
One possibility is that you're actually supposed to use the even extension of
and enforcing a period of
The value of the sum can then be verified by choosing
as required.
