(^592) Series
It follows from Theorem 12.15 that the series 2;pk and Eqk are both con-
vergent. It then follows from Theorem 12.14 that the series E(pk - qk)
and E(pk + qk) are both convergent and hence that the series 2;Uk and
2jukI are both convergent. This completes the proof of Theorem 12.17.
Let Zuk be a series that converges absolutely so that the series Z'Iukl is
convergent. Since the series Zuk is dominated by 1juk!, an application
of Theorem 12.17 (the comparison test) gives the following nontrivial
theorem.
Theorem 12.18 If a series converges absolutely, then it converges.
From the point of view of ordinary elementary mathematics, abso-
lutely convergent series are the ones most easily manipulated. Series
that converge but do not converge absolutely are quite respectable but
can be troublesome. In this course, we learn relatively little about
divergent series and (except for a brief excursion in Problems 5.69) we
never assign values to them. From our present point of view, the asser-
tion "auk = oc " does not mean that the series has a value; it means that
the terms are nonnegative and that the series has partial sums sl, s2, s3,
for which lim s = -.
Problems 12.19
1 Tell the meaning of the statement0=0+0+0+0+
and prove the statement.
2 Using the approximations
log 2 = 0.693, 7r/4 = 0.785, e = 2.71, e'1 = 0.37,draw the interval 0 < x _<_ 1 on a rather large scale and mark the points whose
coordinates are the partial sums st, s2, of the series in the formula(a) 0 =0+0+0+0+0+ -
(b) 1 = + -r - r g -i- 'JIV +
(c) log 2 = 1 +$ 1 +$ +
(d) r/4= 1 -+a+W rr+(e) a-2z +3 +I+Si-} 6i+ +
(I) e=1-11+21_3!+4,_5,+3 Sketch figures indicating the natures of the partial sums of the series(b) 1-2+3-4+5-6+7-8+
(c) 1+0+2+0+3+0+4+0+
(d) 1-1+2-2+3-3+4-4+
(c) (1 -1)+(2-2)+(3-3)+(4-4)+ ...