Motivated by this Math Stack Exchange problem the goal is to evaluate the sum

for any non-negative integer . From looking at this, it is very reasonable to expect that the answer should be a linear combination of the Stieltjes Constants, for , since by definition

One way to evaluate the original sum is to the consider , a partial sum, and repeatedly split things up with the goal of obtaining the Stieltjes Constants. Here however, we will explore a different approach.

Recall the Dirichlet Eta Function which is given by

Then the derivative evaluated at for is equal to the desired sum, that is

Now, with this in mind lets find the expansion of around . Since

and

by multiplying the two series we have that

Consequently, our original sum is the coefficient above multiplied by . Specifically, we have that

For example, when , we have that

and similarly, when ,

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## About Eric Naslund

I am a PhD candidate in mathematics at Princeton University.

Hey it’s not Steilgies, its Steiljes.

Hey thanks Chandru! I see you have a pretty nice blog going there! I am just getting started up (but I have been posting enough)

Hi-

Thanks for your nice opinion. I would like to inform you that there is this link called http://math.is/ which supports MathJaX as a plugin and is very similar to wordpress. You may like to use that site.

Chandrasekhar