12 Series
12.1 Definitions and basic theorems Even though we have already
had experiences with series, we start ab initio to develop the subject.
An array of numbers and plus signs of the form
(12.11) ul+us+u3+
is called a simple infinite series or simply a series. The numbers si, s2,
s3,... defined by sl = u1, s2 = u1 + U2, S3 = ul + u2 + u3, and,
in general,
n
(12.12) s = E uk (n = 1, 2, 3, ...),
k=1
constitute the sequence of partial sums of the series. If it happens that
(12.121) lim S. = s,
n--
then the series is said to be convergent and is said to converge to s. If the
limit does not exist, the series is said to be divergent. A series which
converges to s is not the number s, just as a hand that contains 5 fingers
587