Series question

1. Nov 18, 2007

azatkgz

1. The problem statement, all variables and given/known data
Determine whether the series converges or diverges.

$$\sum_{n=1}^{\infty}\log_{b^n}\left(1+\frac{\sqrt[n]{a}}{n}\right)$$
where a,b>0 some parameters.

3. The attempt at a solution

$$\sum_{n=1}^{\infty}\frac{\ln \left(1+\frac{\sqrt[n]{a}}{n}\right)}{\ln b^n }=\sum_{n=1}^{\infty}\frac{\left(\frac{\sqrt[n]{a}}{n}-O\left(\frac{a^{\frac{2}{n}}}{n^2}\right)\right)}{n\ln b}}$$

$$=\sum_{n=1}^{\infty}\frac{\sqrt[n]{a}}{n^2\ln b}-\sum_{n=1}^{\infty}O\left(\frac{a^{\frac{2}{n}}}{n^3\ln b}\right)$$

So my solution is series converges.

2. Nov 18, 2007

Gib Z

Thats how I would do it as well =]

3. Nov 18, 2007

Thank you!!!