Calculus: Analytic Geometry and Calculus, with Vectors

336 Functions, graphs, and numbers

If it happens that this sequence of partial sums converges to s, so that

(5.621)

n--

then we say that the series converges to s, and, leaving the significance of

the horrendous operations to be revealed in Problem 6 and Chapter 12,

we say that the series has the sum s and we write

(5.622) s = u1 + U2 + U3 + or s = I uk.

k-1

In case a given series is not convergent, we say that it is divergent. The

series

1-1+1-1+1-1+

is a classic example of a divergent series.

We are now ready to attack decimals. Let dl, d2, d3, be a

sequence each element d of which is one of the 10 digits 0, 1, 2, 3, 4, 5,

6, 7, 8, 9. The array

(5.63) O.d1d2d3 ,

in which the first dot is a decimal point, is then an infinite decimal. We

confine our attention to decimals of this form; presence of a positive

integer before the decimal point causes no difficulties. Just as the left

side of the equation

0.31690 =

0

3

+ 102 + 103 + 10' +

lo0

b

is a remarkably efficient way of abbreviating the right side, so also (5.63)

is a remarkably efficient way of abbreviating the infinite series

(5.631) d, d2 d3

10 + 02+ 03+

Thus the infinite decimal is an infinite series in disguise.

Theorem 5.64 Each infinite decimal 0.d1d2d3.. converges to a

real number s.

If we think it will serve a useful purpose, we can say that the decimal

"represents" the number to which it converges. In any case, we write

(5.641) s = 0.d1d2d3

when the decimal converges to s. To prove the theorem, let s denote

the sum of the first n terms of the series (5.631) so that

sn`10+102+...+10n

and

S. = 0.d1d2 ... dn.