Answer
Convergent
Work Step by Step
Since, we have $\lim\limits_{x \to \infty} \int_1^\infty\dfrac{2}{e^x(1+e^x)}$
Suppose $e^x= p \implies dx=\dfrac{1}{p} dp$
or, $=\lim\limits_{a \to \infty} [\dfrac{2}{p}-\dfrac{2}{p+1}]_e^\infty$
or, $=\lim\limits_{a \to \infty} [2(\dfrac{a}{a+1}-2(\dfrac{e}{e+1})]$
or, $ =-2 \ln (\dfrac{e}{e+1}) \approx 0.63$
Hence, the series is Convergent by the Integral Test
