Let denote the -th Bernoulli number. Prove that

**Solution**

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.

Another solution by pprime can be found at aops.com .