Find all such that the series

converges and compute the sum.

First, we have

Then, applying the ratio test we have

Thus, the series converges for all with which implies . Furthermore, if , then the series converges as the alternating harmonic series. Then, we compute the sum for ,

In border cases it should converge to – can be done in a fashion similar to what was done in 10.17.

Shouldn’t the boundary case be according to equation 10.47?