## Calculus: Early Transcendentals 8th Edition

$\lim\limits_{n \to \infty}|\dfrac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{(n+1)!}{(n+1)^{n}b_{1}b_{2}b_{3}.....b_{n} b_{n+1}}}{\frac{n!}{n^{n}b_{1}b_{2}b_{3}.....b_{n}}}|$ $\lim\limits_{n \to \infty}|\dfrac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{n!(n+1)}{(n+1)^{n}b_{1}b_{2}b_{3}.....b_{n} b_{n+1}}}{\frac{n!}{n^{n}b_{1}b_{2}b_{3}.....b_{n}}}|$ $=\lim\limits_{n \to \infty}(\frac{n}{n+1})^{n}\times \frac{1}{b_{n+1}}$ Since, $\lim\limits_{n \to \infty}{b_{n+1}}=\frac{1}{2}$ $=\lim\limits_{n \to \infty}2(\frac{n}{n+1})^{n}$ $=\frac{2}{e}\approx 0.7\lt 1$ By the ratio test, the series absolutely converges.