Which of the following correctly characterizes the series ?

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

Problem

Which of the following correctly characterizes the series

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

? (a) It converges absolutely by the integral test, since

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

converges to a finite value. (b) It diverges because the factor

(\ln n)^{2}

grows too slowly; the series behaves like the harmonic series. (c) The ratio test gives a limit of

1

, so the series diverges. (d) It converges because its terms are eventually smaller than those of

\displaystyle \frac{1}{n^{1.1}}