Answer
Diverges and $\lim\limits_{n \to \infty} a_n=\dfrac{1}{\sqrt [3] {e}}$
Work Step by Step
Let us consider $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (1-\dfrac{1}{3n})^n$
$\implies \lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (1+\dfrac{(-\dfrac{1}{3})}{n})^n$
Thus, we have $\lim\limits_{n \to \infty} a_n=e^{-(1/3)}=\dfrac{1}{\sqrt [3] {e}}$
Here, we get $\lim\limits_{n \to \infty} a_n \ne 0$
Thus, the Series Diverges and $\lim\limits_{n \to \infty} a_n=\dfrac{1}{\sqrt [3] {e}}$
