Answer
Convergent
Work Step by Step
Since, we have $\lim\limits_{x \to \infty} \int_1^\infty\dfrac{e^x}{1+e^x}$
or, $ \lim\limits_{x \to \infty} \dfrac{e^x}{1+e^x}=\lim\limits_{a \to \infty} [tan^{-1} (e^x)]_e^a$
or, $=\lim\limits_{a \to \infty} [tan^{-1} (a)-tan^{-1} (e)]]$
or, $ =\frac{\pi}{2}-tan^{-1} (e)\approx 0.35$
Hence, the series is Convergent by the n-th term Test of Convergence
