Answer
Converges
Work Step by Step
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, $r=\lim\limits_{k \to \infty}(1-\dfrac{1}{k})^k$
and $\implies \ln L = \lim\limits_{k \to \infty} k \ln (1-\dfrac{1}{k}) \\= \lim\limits_{k \to \infty} \dfrac{(1-\dfrac{1}{k})}{1/k}$
Use L-hospital Rule: $L= \lim\limits_{k \to \infty} \dfrac{\frac{d}{dk}(1-\dfrac{1}{k})}{\frac{d}{dk}(1/k)}=\lim\limits_{k \to \infty} \dfrac{-1}{1-\dfrac{1}{k}}=-1$
Now, $L=\lim\limits_{k \to \infty}(1-\dfrac{1}{k})^{-1}=e^{-1} \lt 1$
Therefore, the series converges by the root test.
