Determine whether the series converges or diverges.

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

Problem

Determine whether the series

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

converges or diverges. (a) It diverges because substituting

u=\ln x

incorrectly gives

\int \! \frac{1}{u}\,du

, which is unbounded. (b) It diverges because the logarithm grows too slowly, so the terms behave like the harmonic series

1/n

. (c) The series converges because

\displaystyle\int_{2}^{\infty}\frac{1}{x(\ln x)^{3}}\,dx

is finite; by the Integral Test, the series therefore converges. (d) It diverges; the Ratio Test gives a limit of

1

, which demonstrates divergence.