Calculus: Analytic Geometry and Calculus, with Vectors

5.6 Sequences, series, and decimals 337

The set S consisting of the numbers s1, s2, s3, is then nonempty and
has the upper bound 1, since sn < 1 for each n. Therefore, Theorem 5.46
implies that S has a least upper bound which we denote by s. Then
s < s for each n. To each e > 0 there correspondsan index N such that
s,v > s - e, since otherwise s - e would be an upper bound of S less

than s. But the numbers d1, d2, d3, are all nonnegative, and hence
s - e < s5 s when n > N. Therefore, lim s,,, = s or s = 0.d1d2d3

and Theorem 5.64 is proved. For future reference, we note that

very minor modifications of this proof yield proofs of the following two
theorems.
Theorem 5.65 If the terms of the series ul + u2 +u3 + are
nonnegative and if the sequence of partial sums has an upper bound, then
the series is convergent.
Theorem 5.651 If a sequence s1, s2, s3, is monotone increasing
(that is, sm < sn when in < n) and bounded above (that is, sn < M for each
n) then the sequence is convergent. Similarly, each monotone-decreasing
sequence which is bounded below must be convergent.

In connection with Theorem 5.64, it is often necessary to recognize the
awkward fact that two different infinite decimals can converge to the
same number. For example,

4

= 0.250000 , 4 = 0.249999

This situation occurs, however, only when one of the decimals has only
nines from some place onward. In Theorem 5.64 we started with a deci-
mal and found that it converges to a number. The next theorem is
different; we start with a number and find a decimal which converges to it.
Theorem 5.66 If s is a number for which 0 < s < 1, then there is a
decimal 0.d1d2d3.. which converges to it.
Our proof of this theorem involves manipulation similar to the manip-
ulations of Problem 18 of Problems 5.49, where more details are given.

Let d1 be the greatest integer for which 0.d1 c s. Then s - 0.1 <

0.d, <_ s. Let d2 be the greatest integer for which 0.d1d2 S s. Then
s - 0.12 < 0.d1d2 < s. Let d3 be the greatest integer for which 0.didsd3
s. Then s - 0.11 < 0.did2d3 S s. Continuation of this procedure
yields a decimal 0.d1d2d3. that converges to s so that

s = 0.d1d2d3 ....

We conclude this section with a study of geometric series and repeating
decimals. When x 0 1, the identity

(5.67)

1 -xn

-}- x -f- x2 .+. + xn-1

can be proved either by long division or by multiplying by 1 - x. When

jxj < 1, the sequence jxj, Ixi2, 1xi3, is monotone decreasing and