- #1
azatkgz
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Homework Statement
Determine whether the series converges or diverges.
[tex]\sum_{n=1}^{\infty}\log_{b^n}\left(1+\frac{\sqrt[n]{a}}{n}\right)[/tex]
where a,b>0 some parameters.
The Attempt at a Solution
[tex]\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}}[/tex]
[tex]=\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)[/tex]
So my solution is series converges.