Answer
Diverges
Work Step by Step
Consider $a_n=(-ln (e^2 +\dfrac{1}{n}))^{n+1}$
By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
$l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|(-ln (e^2 +\dfrac{1}{n}))^{n+1}|}=\lim\limits_{n \to \infty} (-ln (e^2 +\dfrac{1}{n}))^{1+\dfrac{1}{n}}=\ln (e^2+0)^{1+0}$
or, $\ln e^2= 2$
so, $l \gt 1$
Thus , the given series Diverges by the Root Test.