What are the values of k that make the series convergent?
Just a thought, give this a try: Let $\displaystyle u = 2 + e^{2x}$. Then we have that $\displaystyle e^x = \sqrt{u - 2}$.
Then your sum becomes
$\displaystyle \sum_{k = 1}^{\infty} \dfrac{\sqrt{u - 2}}{u^k}$
which looks a lot simpler.
-Dan
by plugging random values of k, I observed that the sum increases very very slightly like it will reach finite value for k > 0.5.
If k <= 0.5, the sum will increase gradually to infinite.
Does that mean the series is converges when k > 0.5?
How to prove that mathematically?