Answer
The series $\Sigma_{n=1}^{\infty} \frac{n}{2n+1}$ diverges.
Work Step by Step
We have
$$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}\frac{n}{2n+1}=\lim\limits_{n \to \infty}\frac{1}{2 +1/n}=1/2\ne 0$$
Hence the series $\Sigma_{n=1}^{\infty} \frac{n}{2n+1}$ diverges.
