Mathematics

Problem

Consider the positive term series with general term

a_n = \dfrac{1}{n\,(\ln n)^{p}}

for

n \ge 2

, where

p

is a real parameter. Throughout this question, all logarithms are natural logarithms. (a) Evaluate the improper integral

\displaystyle I(p)=\int_{2}^{\infty} \frac{1}{x\,(\ln x)^{p}}\,dx

in terms of

p

, showing all steps in your calculation. (b) For which real values of

p

does the integral

I(p)

converge? Provide a clear justification based on your result from part (a). (c) Using your work in part (a), determine all values of

p

for which the series

\displaystyle \sum_{n=2}^{\infty} a_n

converges. Explain your reasoning.