Answer
Converges
Work Step by Step
Let us consider $a_n=\ln (1+\dfrac{1}{n})$ and $u_n$ is positive for all the values of $n$.
Here, $f(n)=\ln (1+\dfrac{1}{n})$; $f'(n)=\dfrac{-1}{n(1+n)} \leq 0$
The negative sign shows that, the sequence $u_n$ is decreasing.
Now, $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty}\ln (1+\dfrac{1}{n}) \implies \ln (1+0)=0$
Hence, the series converges by the Alternating Series Test.