Answer
Converges
Work Step by Step
We are given that $a_{n+1}=\dfrac{\sqrt[n] {n}}{2}a_n$ and $a_1=5$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{\sqrt[n] {n}}{2}a_n}{a_n}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{(n)^{n/2}}{2}|$
So, $l=\dfrac{1}{2} \lt 1$
Hence, the series Converges by the ratio test.