Determine the convergence of the series .

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Determine the convergence of the series

\displaystyle \sum_{n=1}^{\infty} \frac{(n+1)!}{n!\,n^{2}}

. (a) The series diverges because it ultimately behaves like

\dfrac{1}{n}

. (b) The series converges because it ultimately behaves like

\dfrac{1}{n^{2}}

. (c) The series converges by the Ratio Test since the ratio of successive terms is less than

1

.},{ (d) The series diverges because the terms do not approach

0

; the factorial in the numerator dominates the denominator.