Answer
divergent
Work Step by Step
Given
$$ \sum_{n=1}^{\infty} \frac{e^n+1}{ne^n+1}$$
Use the Limit Comparison Test with $a_{n}=\dfrac{e^n+1}{ne^n+1}$ and $b_{n}=\dfrac{1}{ n}$
\begin{align*}
\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}&=\lim _{n \rightarrow \infty} \frac{ne^n+n}{ne^n+1} \ \ \text{L'Hopital Rule }\\
&=\lim _{n \rightarrow \infty} \frac{e^n+e^nn+1}{e^nn+e^n} \ \ \text{L'Hopital Rule }\\
&=\lim _{n\to \infty \:}\left(\frac{e^nn+2e^n}{e^nn+2e^n}\right)\\
&=1
\end{align*}
since $\displaystyle\sum_{n=1}^{\infty} \frac{1}{ n }$ is divergent $(p-\text {series } p=1),$ then $\displaystyle\sum_{n=1}^{\infty} \frac{e^n+1}{ne^n+1}$ also divergent