Answer
The series diverges
Work Step by Step
$if\lim _{n\rightarrow \infty }a_{n}\neq 0\Rightarrow \sum ^{n}_{1}a_{n}$ diverges
$\lim _{n\rightarrow \infty }\dfrac {2^{n}+1}{2^{n+1}}=\dfrac {1}{2}+\dfrac {1}{2^{n+1}}=\dfrac {1}{2}+0=\dfrac {1}{2}\neq 0$
So the series diverges