Solving Series Question: Determine Convergence/Divergence

[azatkgz]

Homework Statement

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.

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.

 

[Gib Z]

Thats how I would do it as well =]

 

[azatkgz]

Thank you!