$\displaystyle \displaystyle \begin{align*} \frac{2^n + 1}{2^{n+1}} &= \frac{2^n}{2^{n+1}} + \frac{1}{2^{n+1}} \\ &= \frac{1}{2} + \frac{1}{2^{n+1}} \\ &= \frac{1}{2} + \left( \frac{1}{2} \right) ^{n+1} \end{align*}$
So the series must be divergent because constantly adding $\displaystyle \displaystyle \begin{align*} \frac{1}{2} \end{align*}$ will eventually lead to $\displaystyle \displaystyle \begin{align*} \infty \end{align*}$.