sum of infinte series with exponential and factorial terms Want to sum the following series:
$$
\sum_{t=1}^\infty e^{-tk} \frac{(tk)^t}{t!}
$$
where $k$ is an integer $>0$.

The solution is to estimate:
$$\sum_{t=1}^\infty e^{-tk} \frac{(tk)^t}{t!}$$
Taylor Series rough estimation
$$ f(k) = \sum_{t=1}^\infty e^{-tk} \frac{(tk)^t}{t!}= k + e^{-2k}\frac{2k}{2!} +e^{-3k} \frac{3k^3}{3!}+ e^{-4k} \frac{4k^4}{4!} + e^{-5k} \frac{5k^5}{5!}+ e^{-6k} \frac{6k^6}{6!}$$
if integer $$K > 0$$
