Answer
Converges
Work Step by Step
Here, we have $|a_n|=\Sigma_{n=1}^{\infty} (\dfrac{n}{(\ln n)^n} $
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $r=\lim\limits_{n \to \infty}\sqrt[n] {|a_n|}$
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, $r=\lim\limits_{n \to \infty}\sqrt[n] {(\dfrac{n}{(\ln n)^n} }\\=\lim\limits_{n \to \infty} \dfrac{n^{1/n}}{\ln n} \\=\lim\limits_{n \to \infty} \dfrac{1/n n^{1/n-1}}{1/n}\\=0 \lt 1$
Therefore, the given series converges by the root test.
