Answer
Diverges
Work Step by Step
We have: $a_k=\dfrac{1}{1+\ln k}$ and $b_k=\dfrac{1}{k}$
Now, we need to apply the limit comparison test.
$L=\lim\limits_{k \to \infty}\dfrac{a_k}{b_k}\\=\lim\limits_{k \to \infty}\dfrac{1/1+\ln k}{1\ k}\\=\lim\limits_{k \to \infty} \dfrac{k}{1+\ln k}\\=\infty$
We see that $ \lim\limits_{k \to \infty}\dfrac{a_k}{b_k}= \infty$
and the series $\Sigma b_k$ is a divergent harmonic series.Therefore, the given series diverges by the limit comparison test.
