## A Few More Sums Involving the Zeta Function

**Point of post: **In this post we compute the value of

The reason we do this particular sum is that while the mathworld.com site has many identities relating to the zeta function even a lot that look like this sum, they don’t have this one. It is slightly larder than the one’s on the site, so maybe this is the reason.

**Problem: **Compute

**Solution: **Recall that for all one has

And thus for we have that

And thus, for we may conclude that

But, taking the limit as on both sides gives

where we’ve used the fact that the series on the left had radius of convergence and the series converged at and thus Abel’s Limit Theorem applied. Note then that

where

Letting in this integral gives

Let in this integral to get that

and thus solving we find that

Letting gives us that

Note though that is symmetric on and thus

or letting gives us

Thus,

where we’ve used the fact that . Let in this last integral to get that

Solving we find that

and thus

**References:**

1. Andrews, George E., Richard Askey, and Ranjan Roy. *Special Functions*. Cambridge [u.a.: Cambridge Univ., 2006. Print.

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