Question

Determine if the series is absolutely convergent, conditionally convergent, or divergent.
\[\sum_{2}^{\infty}n \div \left( \ln n \right)^{n}\]

Answer 1

Absolutely convergent...use root test

Answer 2

how do u use the root test, I'm still confused about it

Answer 3

\begin{eqnarray*}\limsup_{n\to\infty}\left(\sqrt[n]{\frac{n}{\log^n(n)}}\right)&=&\limsup_{n\to\infty}\left(\frac{\sqrt[n]{n}}{\log(n)}\right)\\&=&\lim_{n\to\infty}\left(\frac{\sqrt[n]{n}}{\log(n)}\right)\\&=& 0.\end{eqnarray*}Because this number is less than 1, the series is absolutely convergent.