Answer
Converges
Work Step by Step
Given: $a_{n+1}=\dfrac{1+\ln n}{n}a_n$ and $a_1=1$
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{(\dfrac{1+\ln n}{n})a_n}{a_n}|$
This implies that
$\lim\limits_{n \to \infty}|\dfrac{1+\ln n}{n}|=|\lim\limits_{n \to \infty}(\dfrac{1}{n})+\lim\limits_{n \to \infty}(\dfrac{\ln n}{n})|=0 \lt 1$
Thus, the series converges by the ratio test.
