Calculus: Analytic Geometry and Calculus, with Vectors

338 Functions, graphs, and numbers

bounded below by 0 and hence must have a limit. If we let L denote

this limit, then

L = Jim jxjn}1 = Ixl lim Ixln = IxIL,

n--> ae n-+ m

so (1 - Jxj)L = 0 and hence L = 0. Therefore, as we have previously

proved in another way,

(5.671) Jim xn = 0 OxI < 1).

n- -

But the right member of (5.67) is the sum of the first n terms of the series

in the right member of the formula

(5.672) 1 1

x

= 1 + x + x2 + Xa + (1xi < 1).

Hence, when lxi < 1, taking the limits as n becomes infinite of the mem-

bers of (5.67) gives (5.672). Multiplying the members of (5.67) by a
constant a gives the very important formula

(5.673)

a =a+ax+ax2+ax3+...

1 - x (ixI<1)

which must be permanently remembered. The series is a geometric series

with ratio x, the ratio being the factor by which we multiply one term to

get the next. The easy way to remember the formula is to remember

that, when the absolute value of the ratio is less than 1, a geometric series

converges to the first term divided by 1 minus the ratio.

A repeating decimal is one, like

3.16952 952 952

in which, from some place onward, the digits involve only periodic

repetitions of a collection containing one or more digits. With the aid of

(5.673) we can show that each repeating decimal converges to (or is) a

rational number, that is, a quotient of two integers. For example, if s

is the number to which the decimal displayed above converges, then

s

100 +

101

0 (.952 +

190052

0 +(1000)2

316 1 .952 316 1 952

100 + 1001 1--sa 100 +100 999

It is presumed that we can add fractions when there is a reason for doing

so, and we can see that s is a quotient of integers with denominator 99900.

The most important fact concerning repeating decimals is set forth in

the following theorem.

Theorem 5.68 The (terminating or nonterminating) decimal expansion

of each rational number is a repeating decimal.