   Chapter 11.1, Problem 90E

Chapter
Section
Textbook Problem

# Let a n ( 1 + 1 n ) n (a) Show that if 0 ≤ a < b, then b n + 1 − a n + 1 b − a < ( n + 1 ) 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 {an} is increasing. (d) Use a = 1 and b = 1 + 1/(2n) in part (b) to show that a2n < 4. (e) Use parts (c) and (d) to show that an < 4 for all n. (f) Use Theorem 12 to show that limn→∞ (1 + 1/n)n exists. (The limit is e. See Equation 6.4.9 or 6.4*.9.)

(a)

To determine

To prove: If 0a<b , then bn+1an+1ba<(n+1)bn .

Explanation

Proof:

Note that, bn+1an+1=(ba)(bn+bn1a+bn2a2+bn3a3++ban1+an) .

Divide by (ba) on both the sides of the equation.

bn+1an+1(ba)=(ba)(bn+bn1a+bn2a2+bn3a3++ban1+an)(ba)=(ba)(ba)(bn+bn1a+bn2a2+bn3a3++ban1+an)=bn+bn1a+bn2a2+bn3a3

(b)

To determine

To deduce: bn[(n+1)annb]<an+1 .

(c)

To determine

To show: The sequence {an} is increasing.

(d)

To determine

To show: The sequence a2n<4 .

(e)

To determine

To show: an<4 for all n by using parts (c) and (d).

(f)

To determine

To show: The limit of the sequence an=(1+1n)n exists by using Theorem 12.

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