(^590) Series
Theorem 12.15 If Euk is a series of nonnegative terms, so that uk > 0
for each k, and if the sequence sl, s2, of partial sums is bounded, so
that, for some constant M,
(12.151) S. = U1 + u2 + + un < M (n = 1, 2, 3, ),
then y,uk converges.
This theorem was proved in Section 5.6, but the theorem is so important
that we think about it some more. The hypothesis that u,? 0, u2 > 0,713 >_ 0, implies that s1 = ul > 0, s2 n S1 + u2? sl, s3 = s2 +
u3 >- s2, and, in fact, that(12.152) 0<sl<s2<s3< ... <_sn<M
for each n = 1, 2, 3,.. Since the set E consisting of the numbers
s1, s2j s3, is nonempty and has an upper bound, it must, according
to Theorem 5.46, have a least upper bound which we can call s. Then
s,, < s for each n. Let e > 0. There must be an index N for which
sx > s - E, since otherwise s - E would be an upper bound of the set ofnumbers sl, S2, s3,. and s would not be the least upper bound. It
follows from (12.152) that(12.153) :! 9 :! 9 (n>N),
and hence that lim sn = s. Therefore Mun converges to s and Theorem
12.15 is proved. In case the terms of a series Euk are all positive and
s < M for each n, the figure obtained by plotting the partial sums s1,
S2, s3, '. ,the upper bound M, and the number s to which Muk con-
verges must look essentially like Figure 12.154. On the other hand, if0 51 S2 S3 S4 S M
Figure 12.1542uk is a series of nonnegative terms for which the sequence of partial
sums is not bounded, then s.---> co as n--+ x and the series diverges.
For series of nonnegative terms, and for such series only, it is convenient
to use the first of the formulaszuk < 00, Euk = 00to abbreviate the statement that the series is convergent and to use the
second of the formulas to abbreviate the statement that the series is
divergent. In particular, a series Euk is said to converge absolutely, or
to be absolutely convergent, if Mjukl < -o.
Before stating the next theorem, we look at the two seriesao-+alx-+aax2+asx3+ ... 1 +x+x2 +x3+..