## Thomas' Calculus 13th Edition

Apply Root Test such as: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ which states that the series converges when $L \lt 1$; the series diverges when $L \gt 1$ Let us consider $a_n=\dfrac{(n^n)}{(2^n)^2}$ $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} (|\dfrac{(n^n)}{(2^n)^2}|)^{1/n}$ $\implies L=\lim\limits_{n \to \infty} \dfrac{n}{4}=\lim\limits_{n \to \infty} \dfrac{1}{\dfrac{4}{n}}= \dfrac{1}{0}=\infty \gt 1$ Hence, the series diverges by the Root Test.