Answer
The series converges.
Work Step by Step
We are given that $a_k=ke^k=\dfrac{k}{e^k}$
Ratio Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\dfrac{a_{k+1}}{a_k}$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{k \to \infty}\dfrac{(k+1)e^{k}}{k e^{(k+1)}}\\=\lim\limits_{k \to \infty} \dfrac{(k+1) e^k}{k (e)(e^k)}\\=\lim\limits_{k \to \infty} \dfrac{1+\dfrac{1}{k}}{e}\\=\dfrac{1}{e}$
Therefore, the series converges by the ratio test.
