## Interesting Series Involving the Harmonic Numbers

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

where is the harmonic number.

**Problem: **Compute

**Proof: **We begin by recalling that if and are two complex power series convergent on then

converges on and its value is

(this is just a consequence of Merten’s Theorem). In particular, if the power series is real with

on then

Thus, noting that for

we may conclude that

Thus, it follows that

and thus

for . In particular, it follows that

Thus, it suffices to compute

But, letting we find that

Recall though that for we have that

and thus, in particular

But, this is just where is the Dirichlet Eta Function. But, it’s a well-known fact that

where is the Riemann Zeta Function. Thus, it follows that

**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.

[…] begin now by recalling that in our last post we derived the result […]

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