Answer
Diverges
Work Step by Step
The sum of a geometric series can be found as:
$S=\dfrac{a}{1-r}$
Thus, we have $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} \ln ( \dfrac{n}{2n+1})$
and $\lim\limits_{n \to \infty} \ln ( \dfrac{n}{2n+1})=\ln \dfrac{1}{2} \ne 0$
Hence, the given series diverges as per the nth term integral test.
