Test the following series for convergence or divergence. Justify the decision.
Then, the series converges by Example #1 on page 398 of Apostol,
where (since ). Then consider the limit,
The limits of each of the terms in the product exist (as we show below) so the limit of the product is the product of the limits,
The limit since we know
for all , . Therefore, we have
By Theorem 10.9 (see the note after the proof of the theorem on page 396 of Apostol), we then have the convergence of implies the convergence of . Hence,