Answer
Divergent
Work Step by Step
Use Limit Comparison Test with $a_n = \frac{e^n+1}{ne^n+1}$ \ and \ $b_n = \frac{1}{n}$
\\
$\lim\limits_{n \to \infty}\frac{a_n}{b_n}=\lim\limits_{n \to \infty}\frac{e^n+1}{e^n+1}$
\\
Use L'Hopital rule\\
$\lim\limits_{n \to \infty}\frac{e^n}{e^n}=\frac{\infty}{\infty}$
\\
$\sum_{n=1}^{\infty} \frac{1}{n} $is divergent because a p−series with $p=1\leq{1}$ is divergent; thus the series $\sum_{n=1}^{\infty} \frac{e^n+1}{ne^n+1}$ is also divergent
