Answer
Converges
Work Step by Step
Given: $a_{n+1}=(\dfrac{\sqrt[n] {n}}{2})a_n$ and $a_1=5$
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{(\dfrac{\sqrt[n] {n}}{2})a_n}{a_n}|$
$\lim\limits_{n \to \infty}|\dfrac{(n)^{(n/2)}}{2}|=\dfrac{1}{2} \lt 1$
Therefore, the series converges by the ratio test.