finding the sum of a series
It's a very interesting question. There are various techniques for solving such. For example,
This can be solved by using Fourier coefficients and Parseval's Theorem.
is shown by considering:
and differentiating and considering the Taylor series (thanks Daniel).
is solved by considering the sum:
is solved by considering:
and differentiating both the sum and the expression for the sum of the corresponding geometric series with respect to w.
Tons more I bet. Would be nice to have a compilation of the various methods for calculating infinite sums.