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# A Bernoulli series

Let denote the -th Bernoulli number. Prove that

Solution

We begin by the known fact that

It is also well known that the generating function of the Bernoulli numbers is

A simple observation shows that

We have thus establised the identity

(1)

Setting we get that

(2)

Integrating from to and taking limit as we get the result.

By the way, it also holds that

however the above series is not elementary.

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