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# Harmonic sum with reciprocal central binomial coefficient

Let denote the – th harmonic number. Prove that

where denotes the Catalan’s constant.

Solution

We begin with a lemma:

Lemma: Let denote the dilogarithm function. It holds that

Proof: It is well known that

(1)

Setting we have that:

Setting we have that:

However,

where is the inverse tangent function. It now follows that

in view of the well known series .

Substracting the above relations we get the proof of the lemma.

Theorem: Let . It holds that

Setting we have that . The result now follows immediately using the lemma above as well as the fact that .

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