## An Interesting Sum Involving Riemann’s Zeta Function

**Point of post: **In this post we shoe precisely we compute the radius of convergence of the the power series

and find an explicit formula (in terms of known special functions and constants) for values where it converges.

**Theorem: ***The power series *

*converges absolutely on and conditionally at ; it diverges for every . Moreover, for we have that *

*where is the Euler-Maschernoi constant.*

**Proof: **We recall that the radius of convergence of our power series is

Recall though that for integer we have that

so that

Thus, applying the squeeze theorem to both sides gives us that

and thus inputting this into the definition of radius of convergence gives us that . Thus, it follows that

converges for and diverges for . Thus, it remains to check the cases when . Using we readily attain that

from where

converges by Dirichlet’s test. Just as easily from we obtain that

and thus

diverges with comparison the harmonic series. Thus, we have the first part of our result, namely that

converges precisely when . Thus, we are now trying to prove that for such we have that

To do this we first recall that for we have

Then, taking the logarithm of both sides (recalling the result of taking the logarithm of an infinite convergent series of non-negative terms)$ we get

or, said differently

The problem is finished then by noticing that

where the interchanging of the sums was justified since for each and each fixed the inner sum converges absolutely.

**References:**

1. Apostol, Tom M. *Mathematical Analysis.* London: Addison Wesley Longman, 1974. Print.

2. Edwards, Harold M. *Riemann’s Zeta Function*. Mineola, NY: Dover Publications, 2001. Print.

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