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 .