## Thomas' Calculus 13th Edition

Given: $a_{n+1}=\dfrac{1+\tan^{-1} (n)}{n}a_n$ and $a_1=1$ $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{1+\tan^{-1} (n)}{n}a_n}{a_n}|$ $\implies \lim\limits_{n \to \infty}|\dfrac{1+\tan^{-1} (n)}{n}|=|\lim\limits_{n \to \infty}(\dfrac{1}{n})+\lim\limits_{n \to \infty}(\dfrac{1}{n \tan n})|=0 \lt 1$ Thus, the series converges by the ratio test.