Consider the series
Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.
The given series converges absolutely.
Proof. First, we have
for all positive integers . (Since
for
.) Therefore,
Then, we write the Taylor expansion of ,
Thus,
But,
for since
Furthermore, . Hence,
therefore, by the comparison test
converges. So, finally, we have
converges. This implies
is absolutely convergent