Mathematics

Problem

Consider the series

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

. Which statement correctly describes its behavior? a) The series diverges because

\ln n

eventually dominates

n^{3}

. b) The series converges since

\displaystyle\frac{\ln n}{n^{3}}\le\frac1{n^{2}}

for all

n\ge2

. c) The series converges conditionally but not absolutely. d) The series diverges by comparison with

\displaystyle\sum_{n=1}^{\infty}\frac1n