Let denote the -th harmonic number. Prove that forall it holds that

**Solution**

Well we are stating two lemmata.

**Lemma 1: **For all it holds that

*Proof: *Pretty straight forward calculations show that

and Lemma 1 is proved.

**Lemma 2: **For all it holds that

*Proof: *We begin by lemma 1 and successively we have

and Lemma 2 follows.

Now, mapping back at Lemma we have that

Integrating we have that

The interested reader can find at aops.com interesting things about the “root of unity filter” which provides another way of computing the generating function of .