Mathematics

Problem

Consider the improper integral

J(q)=\displaystyle \int_{2}^{\infty} \frac{\ln x}{x^{q}}\,dx

, where

q

is a real parameter. (a) Determine all values of

q

for which

J(q)

converges. Fully justify your answer by analyzing the limit involved in the definition of the improper integral. (b) For every

q

in the interval of convergence obtained in part (a), evaluate

J(q)

in closed form. (c) Compute

\displaystyle \lim_{q\to 1^{+}} J(q)

, and interpret the result in the context of your answer to part (a).