Answer
Converges
Work Step by Step
Consider $u_n=\ln (1+\dfrac{1}{n})$
Here, $u_n$ is positive for all the values of $n$.
Now, $f(n)=\ln (1+\dfrac{1}{n})$
and $f'(n)=\dfrac{-1}{n(1+n)} \leq 0$
The negative sign shows that, the sequence $u_n$ is not increasing.
Thus, $\lim\limits_{n \to \infty} u_n=\lim\limits_{n \to \infty}\ln (1+\dfrac{1}{n})$
or, $\ln (1+0)=0$
Hence, by the Alternating Series Test, the series converges.