Chapter 7.8, Problem 100E

(a)

To determine

To show: The $\underset{n\to \infty }{\lim }\left(\left(\frac{1}{n}{\displaystyle \sum _{k=1}^n\ln k}\right)-\ln n\right)=\underset{n\to \infty }{\lim }\frac{1}{n}{\displaystyle \sum _{k=1}^n\ln \frac{k}{n}}$.

(b)

To determine

To find: The given improper integral using integration of by parts.