## University Calculus: Early Transcendentals (3rd Edition)

Consider $a_n=\dfrac{1}{n^{1+n}}$ By the Root Test $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ $l=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{1}{n^{1+n}}|}=\lim\limits_{n \to \infty}\dfrac{1}{(n)^{1/n}(n^n)^{1/n}}$ or, $\dfrac{1}{\infty}=0$ so, $l \lt 1$ Thus , the given series Converges by the Root Test.