Answer
Diverges
Work Step by Step
Given $$\sum_{n=1}^{\infty}\frac{1}{(\ln n)^{10}} $$ Since for $n\geq1$ \begin{align*} \ln n &\leq n^{1 / 10}\\ \frac{1}{(\ln n)^{10}} &\geq \frac{1}{\left(n^{1 / 10}\right)^{10}}\\ \frac{1}{(\ln n)^{10}} &\geq \frac{1}{n} \end{align*} Compare with $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$, a divergent $p- $series $ (p=1)$; then $\displaystyle\sum_{n=1}^{\infty}\frac{1}{(\ln n)^{10}} $ also diverges
