Using Residues to Sum an Alternating Series

Suppose we are required to find the exact value ofwhereis an even function. We can replace theactor bywhich has simple poles at the pointswith residue

This is important for summing an alternating series. Consider the functionwhereis an even function analytic onapart from a finite number of poles, none of which occur at an integer apart from 0 possible. The residue ofat a non – zero integeris

We integrate the functionaround the square contourwith corners atwhereis large enough to contain all non zero polesofIt follows from the Residue Theorem that

sinceis even.

Now let n tend to infinity. If %phi is such thattends to 0 astends to infinity then


Example: Find

The functionis even and analytic onapart from simple poles at

and similarly for the pole at

Sinceis analytic at

Iflies onthensofor

so by the Estimation Theorem,as