Answer
Converges
Work Step by Step
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: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$
Let us consider $a_n=(1-\dfrac{1}{n})^{n^2}$
$L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|(1-\dfrac{1}{n})^{n^2}|}$
$\implies \lim\limits_{n \to \infty}\sqrt [n] {|((1-\dfrac{1}{n})^{n})^n|}=\lim\limits_{n \to \infty} (1+(-\dfrac{1}{n}))^{n}=(e^{(-1)})=\dfrac{1}{e} \lt 1$
Hence, the series Converges by the Root Test.
