Test the following series for convergence or divergence. Justify the decision.

Let

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,

converges.

This can be done also simpler (from my point of view): notice that the logarithm term is less than n^{1/8} for all n > some N. Then, sqrt{2n – 1} < sqrt{2n}. From this, we find the identity a_n < \sqrt{2}b_n, which leads to Riemann zeta with power 11/8. Thus, by comparison test the series is convergent.