Prove that the following sum converges and has the given value.

Denote the th term in the sum by , so we have

Then, let

Thus, . Therefore, by Theorem 10.4 (page 386 of Apostol, on the convergence of sums of telescoping series) we know the series converges since the sequence converges. Furthermore, we can evaluate the sum,

Therefore,

Hi, thanks a lot for the work you have done, maybe I’m a little bit of a noob but before the end of the proof there is a 2 before the sum and I don’t know if its necessary