## Thomas' Calculus 13th Edition

Apply Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ This test states that when $L \lt 1$; the series converges and for $L \gt 1$; the series diverges. Let us consider $a_n=\dfrac{n}{(\ln n)^n}$ Here, we have $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{n}{(\ln n)^n}|)^{(1/n)}$ $\implies L=\lim\limits_{n \to \infty} \dfrac{\sqrt [n] n}{ \ln (n)}=\dfrac{1}{\infty}=0 \lt 1$ Hence, the series converges by the Root Test.