1549901369-Elements_of_Real_Analysis__Denlinger_

98 Chapter 2 • Sequences

That is, Xn - Xn+l 2 0, from which it follows that Xn+l :S Xn·

(c) By (a) and (b) together, {xn};;:"= 2 is a monotone decreasing sequence

that is bounded below. Therefore, by the monotone convergence theorem, {xn}

converges.

(d) Let L = lim Xn· Then L 2 0, since Xn > 0 for all n. By taking the

n->OO

a

Xn+-

limit of both sides of the equation Xn+i = --

2

• xn_
  a
  L+-

we have

L = __ L_ or 2L = L + !:_

2 ' L.

That is, L = I , or L^2 = a

and thus, Lis a positive number whose square is a; i.e., L =JO,.

(e) Suppose n 2 2. From the last line of (a) above, we see that Xn 2 JO,.

Thus,

a a

• :S r;; = Va :S Xn, SO
  Xn ya
  0 < X - 'a < X - -a = x_ n^2 _ - a < x n^2 __ - a. Thus


• n Vu - n Xn Xn - Va '
  x^2 - a
  Vn 2 2, 0 :S Xn - Va :S fa. •

*Example 2.5. 12 (Calculating y'a to Any Specified Degree of

Accuracy)

The inequality (8) in Theorem 2.5.11 is very useful in practice, to find JO,
to a specified degree of accuracy. For example, suppose we wish to calculate
v's correctly to six decimal places, using the sequence { Xn} defined in Theorem
2.5.11. We want n such that
0 :S Xn - Va :S .0000005 ( = 5 X 10-^7 ).
By (2) it is sufficient to have
x^2 - 5
nv's < 5 x 10-^7 , for which it suffices to have

x_ n^2 __ - (^5) < 5 X 10- 7
2
x; - 5 < 10-^5
x;.<5+10-^6.