Calculus: Analytic Geometry and Calculus, with Vectors

588 Series

is not the number 5. Nevertheless, we find it very convenient to abbre-
viate the statement that the series converges to s by writing one or the
other of

(12.13) s = u1 + U2 + u3 + S = Z U.

k=i

The significance of this matter is usually not fully comprehended by

unfortunate people who have not digested the contents of Problems

6 and 7 of Problems 5.69. For present purposes, it is important to

recognize that the equality signs in (12.13) do not have meanings like

those of the equality signs in ordinary arithmetic and algebra, and that

we cannot get s by "adding up" all of the terms of the series. Passage

of time may possibly bring extinction to the habit of calling s the sum

of the series. The trouble is that the habit makes the theory of series

seem too easy for quick-witted superficial people, and, at the same time,

seem too mysterious and difficult for everyone else. Keeping the impor-

tance of (12.121) constantly in mind enables us to make rapid progress

with the elementary theory of series.

There are numerous reasons why the geometric series

(12.131) a+ar+ar2+ar3+

is important in advanced as well as elementary mathematics. We

should always be well aware of the fact that if s is the sum of n terms

of this series, then, when r 1,

(12.132) r2 + +r"-1)=a11

-r

If Ird < 1, then lim rn = 0, so lim s = a/(1 - r) and hence

(12.133) 1 a

r

= a + ar + ar2 + ar3 +

If Irl >- 1, the series diverges.

The series

(12.134) 1-1+1-1+1-1+1-1+

(1111 < 1).

in which the terms are alternately +1 and -1, has partial sums

(12.135) 1, 0, 1, 0, 1, 0, 1, 0,.

which are alternately 1 and 0, and we start cultivating a good habit by

plotting the points s, = 1, S2 = 0, s3 = 1, s4 = 0,. as in Figure

12.136. There is clearly no s such that lim s = s, and therefore the

Figure 12.136

(^01) S2.S4,S6..--- l
Sg, S3. SS...