Mathematics

Problem

Consider the series

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

. Which statement best justifies that an improper integral of the related function may be used to study the series? a) For

x\ge 2

f(x)=\dfrac{1}{x(\ln x)^2}

is positive and continuous, therefore the integral test can be applied. b) Because

\displaystyle\lim_{n\to\infty}\dfrac{1}{n(\ln n)^2}=0

, the integral test is valid. c) For

x\ge 2

f(x)=\dfrac{1}{x(\ln x)^2}

is positive, continuous, and decreasing, so the integral test may be used. d) Since

f'(x)=\dfrac{1-2\ln x}{x^2(\ln x)^3}

is positive for

x\ge 2

f

is increasing, and monotonicity alone guarantees the integral test applies.