Determine the behavior of the series .

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

Problem

Determine the behavior of the series

\displaystyle\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^3}

. (a) The series converges conditionally but not absolutely. (b) Because

\ln n

grows slower than any power of

n

, the series behaves like a harmonic series, so it diverges. (c) The series converges, because the associated improper integral from

2

\infty

is finite. (d) Using

\displaystyle\int_{2}^{\infty}\!\frac{dx}{x(\ln x)^3}=\ln(\ln x)\big|_{2}^{\infty}=\infty

, the series must diverge.