Answer
Converges
Work Step by Step
Here, we have $|a_n|=\Sigma_{n=1}^{\infty} (\dfrac{1}{n}-\dfrac{1}{n^2})^n $
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $r=\lim\limits_{n \to \infty}\sqrt[n] {|a_n|}$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{n \to \infty}\sqrt[n] {(\dfrac{1}{n}-\dfrac{1}{n^2})^n}\\=\lim\limits_{n \to \infty} (\dfrac{1}{n}-\dfrac{1}{n^2})^{n/n} \\=0 \lt 1$
Therefore, the given series converges by the root test.
