Answer
Diverges
Work Step by Step
Let us apply the Ratio Test to the given series.
In order to solve this series, we have:
$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} \dfrac{n}{2(n+1)^2}\\ =\lim\limits_{n \to \infty} \dfrac{\dfrac{1}{n}}{2(1+\dfrac{1}{n})^2}\\=\dfrac{0}{2(1+0)^2} \\=0 \lt 1$
Thus, the series Diverges by the ratio test.