Calculus: Analytic Geometry and Calculus, with Vectors

12.1 Definitions and basic theorems 591

in which it is supposed that x is a positive number and ao, a,, a2,

are numbers, not necessarily positive, for which Iakl 5 1 for each k =
0, 1, 2,. Employing the admirable terminology of the following
definition, we can see that the first of the two series is dominated by the

second.

Definition 12.16 The series UI + U2 + u3 + is said to be

dominated by the series M1 + M2 + M3 + if Iuxl < Mk for each

k = 1, 2,.
Observe that the terms of a dominating series must be nonnegative;

the inequality lukl 5 Mx, can never be satisfied unless Mk > 0. The
following theorem is known as the comparison test for convergent series;
it tells us that we can be sure that a given series is convergent if we can
find a convergent series that dominates it.
Theorem 12.17 (comparison test) If the series ul + u2 + U3 +
is dominated by a convergent series M, + M2 + M3 + , then the
dominated series must be convergent and, moreover, must be absolutely

convergent.
This theorem assures us, in slightly different words, that a given series
must be convergent if we can find a series of bigger fellows that is con-

vergent. For the case in which the terms are nonnegative, it assures us
that a given series must be divergent if we can find a series of smaller
fellows that is divergent. Our proof of the theorem depends upon use
of two series Epk and Eqk, with terms defined by

(12.171) nkl + Uk

l ukI - Uk

l ) Pk =^2 f qk =^2

that are useful for other purposes. Observe that pk = uk when Uk >= 0,
that Pk = 0 when uk 5 0, and that

05pk5IukI5Mk

in each case. Observe also that qk = 0 when uk? 0, that qk = -uk
when uk 5 0, and that

05gk5lukl5Mk

in each case. Observe finally that adding and subtracting the formulas
in (12.171) gives

(12.172) uk = PA- qk, Iukl = Pk + qk.11,

Letting M be the number to which the series EMk of nonnegative terms
converges, we see that

PI+P2+ ...+ pn S MI+M2+ ... +M 5 M

gl+g2+ ... +qsMl+M2+ ... +M<M.