Convergence test for: $\sum\limits_{n = 1}^{\infty} \frac{n}{e^n}$ I know, for example, that the series:
$$\sum\limits_{n= 1}^{\infty} \frac{1}{e^n}$$
is a geometric series of the form $\sum\limits_{n = 1}^{\infty} k x^n$, where $k = 1$ and $x = \frac{1}{e}$ and it is convergent1. But when I have, as in my case:
$$\sum\limits_{n = 1}^{\infty} \frac{n}{e^n}$$
a ratio of functions I'm struggling to find a method to find whether the series is convergent/ divergent and to find a value in the former case. Thus the following question arises:
What series convergence method to use in case of ratio of functions?

1. And its value is: $\frac{k}{1 - x} - 0^{th}term$

Let $a_n = \frac{n}{e^n}$. Then by ratio test you get that:
$$ \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n\to\infty} \left| \frac{\frac{n+1}{e^{n+1}}}{\frac{n}{e^n}}\right| = \frac{1}{e} < 1.$$
So you can conclude that the series is convergent. Hope that it helps you :)
