I can show that $\sum^\infty\_{k=1}{{1}\over{k^2}} = {{\pi^2}\over{6}}$ without to much hassle just using two representations of ${{sin(x)}\over{x}}$ but I cant find any proof nearly as simple when I try to determine what $\sum^\infty\_{k=1}{{1}\over{k^2+1}}$ equals. 


Does there exist an elementary proof? or must I resort to using integrals? 



algebraic proof of an infinite sum
You can evaluate a series for cotangent at the square root of -1: see 


<http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions> .


