Mathematics

Problem

Let

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

for

x\ge 2

. The infinite series

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

is defined in terms of the function

f

. Answer the following questions about

f

. (a) Explain why

f(x)>0

for all

x\ge 2

. Clearly identify any interval endpoints that require special attention. (b) Describe the continuity of

f

[2,\infty)

. Be sure to justify your answer in terms of the algebraic form of

f

. (c) Determine on which subintervals of

[2,\infty)

the function

f

is increasing and on which it is decreasing. Support your conclusions with analytic reasoning.