Examine the infinite series and the associated function for .

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Examine the infinite series

\displaystyle T=\sum_{n=2}^{\infty} \dfrac{\ln n}{n}

and the associated function

p(x)=\dfrac{\ln x}{x}

for

x\ge 2

. (a) Explain why

p(x)>0

for all

x\ge 2

. Address any boundary considerations explicitly. (b) Determine analytically an interval

[N,\infty)

on which

p

is decreasing, and state the smallest integer

N

that satisfies this property. (c) Compute the improper integral

\displaystyle \int_{2}^{\infty} \dfrac{\ln x}{x}\,dx

, showing each stage of your calculation. (d) Using the information obtained in parts (a)–(c), state whether the series

T

converges or diverges. Provide a logical justification for your answer.