Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{2 n+1}{4^{n}}$$ Since $$ 2 n+1\lt2^n,\ \ \ \ \ \ \ n \geq 3$$ Then $$\frac{2 n+1}{4^{n}}\lt\frac{2^{n}}{4^{n}}=\left(\frac{1}{2}\right)^{n}$$
Since $\sum \left(\frac{1}{2}\right)^{n}$ is a convergent geometric series, then
$\sum_{n=1}^{\infty} \frac{2 n+1}{4^{n}}$ also converges.