96 Chapter 2 • Sequences
(iii) there is one more term than in the expansion of ( 1 + ~) n
(1 )n+l
Putting (i)- (iii) together, we see that 1 + n +
1
>is, the sequence { ( 1 + ~) n} is strictly increasing.
(b) Next, we show that this sequence is bounded above. From the proof of
Part (a), we see that Vn E N ,
(
l)n 1 1 1 1
(^1) + ~ < (^1) + (^1) + 2 + 2 · 3 + ~ + ... + n!
1 1 1 1
< 1 + 1 + 2 + 2. 2 + ~ + ... + 2 n-l
= 1 + (2 - -
1
2 n--) l (See Exercise 1.3.10.)
< 3.
Hence, this sequence is bounded above by 3.
(c) Finally, by P arts (a) and (b) above, { (i + ~) n} is a monotone in-
creasing sequence with an upper bound. Hence, by the monotone convergence
theorem, this sequence converges. D
Definition 2.5.10 (The Number e) e = lim (i + 2-)n
n-+oo n
EXISTENCE OF SQUARE ROOTS, AND HOW TO FIND THEM
In Chapter 1 we showed that in any complete ordered field there is a positive
number whose square is 2. We can use sequences to extend that result, showing
that every positive number a has a square root. The method we use is based on
an ancient method of calculating square roots, called the "guess-and-average"
method. We shall use it to find a sequence of real numbers converging to fa.
The procedure goes as follows. Pick any a > 0. We want to find fa; that is, a
positive number whose square is a. Let x 1 be any positive real number; it will
serve as our "first guess." As our second guess, we take the average of x 1 and
.i!:.... X1 That is,
(If we are lucky and our first guess is exactly fa, then x 1 = .!!:.... , and x 2 = x 1 .)
X1
We repeat this process over and over to define a sequence { Xn} inductively.
The following theorem guarantees that this sequence converges to a positive
number whose square is a.