1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 95

Part 2- find lim Xn· By P art 1, we know that 3 L = lim X n E R We

n-+oo n-+ao

proceed to find L. Consider the defining equation: Xn+l = Jxn + 2. By squaring

both sides, we have

(xn+1)

2

= Xn + 2. Thus,

lim (xn+1)

2

n-+oo - = n-+lim oo (xn + 2).

Applying the algebra of limits to both sides of this equation,

£^2 = L + 2

£^2 - L- 2 = 0

(L - 2)(£ + 1) = 0

L = 2 or L = -1.

Now, L =/:--1 (see Theorem 2.3.12 (b) with K = 0). Thus, L = 2. D

Example 2.5.9 (The Number e) The sequence { (1 + ~) n} converges.

Proof. (a ) We first show that this sequence is strictly increasing. By the

binomial theorem , ( 1 + ~) n

l+n(~) + n(n-1) (~)

2

+ n(n-l)(n-2) (~)

3

+··· + (~)n

n 2 n 2·3·, n. n

l+n(~)+~~n-l+l~n-ln-2+··+ 1 [~n-ln-2···~] =
n 2n n 2·3n n n 2·3···n n n n n

1+1+~ (1-~) +-

1

(1-~) (1-~) +·. ·+~ [(1-~) (1-~) ...

2 n 2 · 3 n Jl n! n n

(l-n~l)J.

1

;

, ( 1 )n+l

Again, by the binomial theorem , 1 + --

n + 1

1+1 + ~ (1--

1

-) + -

1

• (1--

1

-) (1-·-·-

2

-) + ... +

2 n+l 2 · 3 · n+l n+l

(n~~)! t(^1 ~ n:i) (l-n!1) ··· (l-n:1)]·

(

1 ) n+l

Notice that in the above expansion of 1 + n +

1

,

(i) all t erms are positive;

(ii) each of the first n t erms is greater than the corresponding t erm in the

expansion o'f ( 1 + ~) n ;