Chapter 9.2, Problem 32ES

(a) Compare appropriate areas in the accompanying figure to deduce the following inequalities for $n\ge 2:$

${\displaystyle {\int }_1^n\ln xdx<\ln n!<{\displaystyle {\int }_1^{n+1}\ln xdx}}$

(b) Use the result in part (a) to show that $\frac{n^n}{e^{n-1}}<n!<\frac{{\left(n+1\right)}^{n+1}}{e^n},n>1$

(c) Use the Squeezing Theorem for Sequences (Theorem 9.1.5) and the result in part (b) to show that $\underset{n\to +\infty }{\lim }\frac{\sqrt[n]{n!}}{n}=\frac{1}{e}$