Answer
Divergent
Work Step by Step
Let us consider $a_n=\lim\limits_{n \to \infty} \dfrac{2^n}{n+1}$
Here, $\lim\limits_{n \to \infty} \dfrac{2^n}{n+1}=\dfrac{\infty}{\infty}$
This shows that the limit has an indeterminate form so, we will use L-Hospital's rule.
so, $\lim\limits_{n \to \infty} \dfrac{(2^n) (\ln 2)}{1} =\infty $
Thus, the series is divergent.
