## Thomas' Calculus 13th Edition

Let us consider $a_n=\dfrac{8}{(3+\dfrac{1}{n})^{2n}}$ In order to solve the given series we will take the help of Root Test. This test states that when the limit $L \lt 1$, the series converges and for $L \gt 1$, the series diverges. In order to solve the series we will take the help of Root Test. It can be defined as follows: $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty} |a_n|^{1/n}$ $L=\lim\limits_{n \to \infty} \sqrt [n] {|a_n|}=\lim\limits_{n \to \infty}\sqrt [n] {|\dfrac{8}{(3+\dfrac{1}{n})^{2n}}|}$ $\implies \lim\limits_{n \to \infty} \dfrac{8}{(3+\dfrac{1}{n})^{2}}=\dfrac{8}{(3)(3)}=\dfrac{8}{9} \lt 1$ Hence, the series Converges by the Root Test.