Answer
Diverges
Work Step by Step
Here, we have $a_n= \dfrac{2^n}{n^3}$
Ratio Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $l=|\lim\limits_{n \to \infty}\dfrac{a_{n+1}}{a_n}|$
a) When $0 \leq l \lt 1$, the series converges. (b) When $l \gt 1$, or, $\infty$, so the series diverges. (c) When $l=1$, the ratio test is said to be inconclusive.
Now, $l=\lim\limits_{n \to \infty}|\dfrac{\dfrac{2^{n+1}}{(n+1)^3}}{\dfrac{2^n}{n^3}}|\\=2 \times \lim\limits_{n \to \infty}|\dfrac{n^3}{(n+1)^3}|\\=2$
so, $l \gt 1$
Therefore, the series diverges by the ratio test.
