Answer
Converges
Work Step by Step
We can write the general form of the given series as: $a_k=k e^{-k}$
Root Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\sqrt[k] a_k$
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, $r=\lim\limits_{k \to \infty}\sqrt[k] {k e^{-k}} \\=\lim\limits_{k \to \infty} \dfrac{1}{e} \times \lim\limits_{k \to \infty} k^{1/k} \\=\dfrac{1}{e} \times \lim\limits_{k \to \infty} e^{\ln k/k} \\=(\dfrac{1}{e})(e^0)\\=\dfrac{1}{e} \lt 1$
Therefore, the series Converges by the root test.
