Chapter 11.1, Problem 90E

Let $a_n={\left(1+\frac{1}{n}\right)}^n$.

(a) Show that if $0\le a<b$, then

$\frac{b^{n+1}-a^{n+1}}{b-a}<\left(n+1\right)b^n$

(b) Deduce that $b^n\left[\left(n+1\right)a-nb\right]<a^{n+1}$.

(c) Use $a=1+1/\left(n+1\right)$ and $b=1+1/n$ in part (b) to show that $\left\{a_n\right\}$ is increasing.

(d) Use $a=1$ and $b=1+1/\left(2n\right)$ in part (b) to show that $a_{2n}<4$.

(e) Use parts (c) and (d) to show that $a_n<4$ for all n.

(f) Use Theorem 12 to show that ${{\displaystyle \lim }}_{n\to \infty }{\left(1+1/n\right)}^n$ exists.

(The limit is e. See Equation 6.4.9 or ${6.4}^{*}.9$.