Answer
Diverges
Work Step by Step
The Comparison Test states that the p-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ is convergent if $p\gt 1$ and divergent if $p\leq 1$.
We can use the Comaprsion Test since both series are clearly positive. The left series is always larger than the right because $\ln{k}$ is greater than $1$, the numeraror of $1/k$ , once $k$ reaches $3$.
Therefore,
$\Sigma_{k=1}^{\infty} \frac{lnk}{k}\geq \Sigma_{k=1}^{\infty} \frac{1}{k} $ for $k\geq 3$
Since, $1/k$ ia s p-series with $p\leq 1$ , the series on the right diverges, and therefore, the series on the left diverges.
