Answer
Diverges
Work Step by Step
Apply the ratio test.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty} \dfrac{4^{k+1}}{(k+1)^2} \times \dfrac{k^2}{4^k}\\=\lim\limits_{k \to \infty} \dfrac{4 \cdot 4^k}{4^k} \times (\dfrac{k}{k+1})^{2}\\=\lim\limits_{k \to \infty} 4 (\dfrac{k+1}{k-1}-\dfrac{1}{k+1})^{2}\\=4(1-0)^2 \\=4 \gt 1$
So, we can conclude that the given series diverges by the ratio test.
