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A generating function involving harmonic number of even index

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

1 Comment

1. 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 .

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