Answer
Diverges
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
Now, $\lim\limits_{n \to \infty} s_n=\lim\limits_{n \to \infty} \ln ( \dfrac{n}{2n+1})$
and $\lim\limits_{n \to \infty} \ln ( \dfrac{n}{2n+1})=\ln \dfrac{1}{2}$
Also, $\lim\limits_{n \to \infty} \ln ( \dfrac{n}{2n+1}) \ne 0$
Hence, the series diverges as per Nth Term Integral Test.