Answer
Converges
Work Step by Step
We are given that $a_{n+1}=\dfrac{1+\ln n}{n}a_n$ and $a_1=1$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{1+\ln n}{n}a_n}{a_n}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{1+\ln n}{n}|=\lim\limits_{n \to \infty}|\dfrac{1}{n}+\dfrac{\ln n}{n}|$
So, $l=0 \lt 1$
Hence, the series Converges by the ratio test.