Answer
Converges
Work Step by Step
Let $u_n=\dfrac{2^n}{3+4^n}$ and $v_n=(\dfrac{ 1}{2^n})$
Now, $\lim\limits_{n \to \infty}\dfrac{u_n}{v_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{2^n}{3+4^n}}{ 1/2^n}$
$\implies \lim\limits_{n \to \infty} \dfrac{4^n}{3+4^n}=1$
Thus, the series converges due to the limit comparison test.