Chapter 11.1, Problem 90E

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

(a) Show that if 0 ≤ a < b, then

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

(b) Deduce that bⁿ[(n + 1)a − nb] < aⁿ⁺¹

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

(d) Use a = 1 and b = 1+1/(2n) 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 lim_n→∞ (1 + 1/n)ⁿ exists. (The Limit is e. See Equation 3.6.6.)