Answer
Converges
Work Step by Step
Given
$$\sum_{n=2}^{\infty} \frac{1}{2^n+1}$$
By using the Ratio Test, we get:
\begin{align*}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
&=\lim _{n \rightarrow \infty}\frac{2^n+1}{2^{n+1}+1} \\
&= \lim _{n \rightarrow \infty} \frac{2^{n}\left(1+\frac{1}{2^{n}}\right)}{2^{n}\left(2+\frac{1}{2^{n}}\right)}\\
&= \frac{\lim _{n \rightarrow \infty}\left(1+\frac{1}{2^{n}}\right)}{\lim _{n \rightarrow \infty}\left(2+\frac{1}{2^{n}}\right)}\\
&=\frac{1}{2}<1
\end{align*}
Thus the series converges.
