Answer
Convergent
Work Step by Step
Consider $ \lim\limits_{x \to \infty} \dfrac{e^x}{1+e^x}=\lim\limits_{k \to \infty} [\tan^{-1} (e^x)]_e^k$
and $\lim\limits_{k \to \infty} [\tan^{-1} (k)-\tan^{-1} (e)]=(\dfrac{\pi}{2})-\tan^{-1} (e)\approx 0.35$
As per the n-th term Test of Convergence, the series is Convergent.