Use residues!  For an entertaining narrative with the correct answer, see [here][1].  For a derivation, see a complex analysis text.

### Added
 Here's a link to an outline of the residue method of solving this: [Conway page 122][2]. 


  [1]: https://mathoverflow.net/questions/8741/justifying-a-theory-by-a-seemingly-unrelated-example/8752#8752
  [2]: http://books.google.com/books?id=9LtfZr1snG0C&lpg=PP1&dq=complex%2520analysis%2520conway&pg=PA122#v=onepage&q=&f=true