Problem 90E

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

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

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

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

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

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

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

(f) Use Theorem 12 to show that $\lim _{n \rightarrow \infty}(1+1 / n)^{n}$ exists. (The limit is $e$. See Equation $6.4 .9$ or $6.4^{*} .9 .$ )

Step-by-Step Solution