Answer
Diverges
Work Step by Step
Let $u_n=\dfrac{1}{1+\ln n}$ and $v_n=(\dfrac{1}{n})$
Now, $\lim\limits_{n \to \infty}\dfrac{u_n}{v_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{1}{1+\ln n}}{1/n}$
$\implies \lim\limits_{n \to \infty} \dfrac{n}{1+\ln n}=\infty$
Hence, the series diverges by the limit comparison test.