Do you really mean a series? If you mean the series
Σ eⁿ/(1 + e^(2n)),
then yes it is convergent. If you do a limit comparison test to the convergent geometric series Σ 1/eⁿ, you get
lim (eⁿ/(1 + e^(2n))/(1/eⁿ) = 1.
As this limit is positive and finite, both series converge or both series diverge. The series Σ 1/eⁿ is known to be convergent, so your series is also convergent.
You can also use the integral test, or direct comparison, or some other test.