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?