Answer
Diverges
Work Step by Step
Let us consider $a_n=(-ln (e^2 +\dfrac{1}{n}))^{n+1}$
In order to solve the given series we will take the help of Root Test. This test states that when the limit $L \lt 1$ , the series converges and for $L \gt 1$, the series diverges. In order to solve the series we will take the help of Root Test.
It can be defined as follows: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$. In order to solve the series we will take the help of Root Test.
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|(-ln (e^2 +\dfrac{1}{n}))^{n+1}|}$
$\implies \lim\limits_{n \to \infty} (-ln (e^2 +\dfrac{1}{n}))^{1+\dfrac{1}{n}}=\ln (e^2+0)^{1+0}=\ln (e^2)= 2 \gt 1$
Hence, the series Diverges by the Root Test.