Answer
Diverges
Work Step by Step
We can write the general form of the given series as: $a_k=\dfrac{2^k}{k^4}$
Ratio Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\dfrac{a_{k+1}}{a_k}$
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, $r=\lim\limits_{k \to \infty}\dfrac{2^{k+1} (k^4)}{2^k (k+1)^4} \\= \lim\limits_{k \to \infty} \dfrac{2 \cdot k^4}{(k+1)^4} \\=2$
Therefore, the series diverges by the ratio test.
