Find if the series converges or diverges: $ a_n=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^2}\right)^n $
Simplifying the series expression we get $ \left(\frac{n-1}{n^2}\right)^n=\frac{\left(1+\frac{-1}{n}\right)^n}{(n)^{2n}}, $ conducting Root test, taking $n$-th root of the simplified expression as $n \to \infty$, $e^{-1}$. Is this methods correct? OR as the author has done by taking the $n^{th}$ root of the original expression of $a_n$, we get $ \lim_{n\to\infty}\left(\frac{1}{n}-\frac{1}{n^2}\right) =0 \Rightarrow a_n $ converges?