## Another Interesting Series Involving the Harmonic Numbers

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

and in doing so also calculate

where is the harmonic number.

**Problem: **Compute

**Proof:**

**Lemma: **Let . Then,

where is the Gamma Function and is the Riemann Zeta Function.

**Proof: **Recall by definition that

We note then that

where the second step was due to the substitution . It follows then that

where we’ve used tacit use of the uniform continuity a.e. for the summand and the geometric series.

** **We begin now by recalling that in our last post we derived the result that

for . It follows then that for we have that

and thus for we have that

Noting though that since this series on the right converges for since (where is the Euler-Mascheroni constant) it follows that

Note though that if one lets in this integral one arrives at

But, by our lemma this is equal to

The problem follows since

**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.

[…] was browsing thorough Alex Youcis’s blog Abstract Nonsense, when I stumbled upon a post where I saw the […]

Pingback by Zeta function and harmonic numbers « Hardy-Ramanujan Letters | March 14, 2011 |

Hello Alex,

Interesting identity. Based on this idea, I was ale to derive a few more identities involving the harmonic number and the Riemann zeta function. I have mentioned about you in my post

http://hardyramanujan.wordpress.com/2011/03/14/zeta-function-and-harmonic-numbers/

Comment by Nilotpal Sinha | March 14, 2011 |

Dear Niloptal,

Thanks for visiting my blog! Out of all my posts, this is not one I would have thought someone would enjoy. Also, I have never seen your blog, but it is very, very impressive. It is definitely of a much different taste than what I am used to–all the better! Keep up the good work friend.

Best,

Alex Youcis

Comment by drexel28 | March 15, 2011 |