Answer
Divergent
Work Step by Step
Given: $\Sigma_{n =1}^{\infty} \dfrac{e^n}{e^n+n}$
Thus, we have:
$\lim\limits_{n \to \infty} \dfrac{e^n}{e^n+n}= \dfrac{\lim\limits_{n \to \infty} 1}{\lim\limits_{n \to \infty}(1+\dfrac{n}{e^n})}=\dfrac{\lim\limits_{n \to \infty} 1}{\lim\limits_{n \to \infty} 1+ \lim\limits_{n \to \infty}\dfrac{n}{e^n}}=\dfrac{1}{1+0}=1\ne 0$
Hence, the series is divergent as per the nth-term integral test.