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.
Lemma: Let . Then,
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
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.