1550251515-Classical_Complex_Analysis__Gonzalez_

190 Chapter4

This example also shows that the "dissociative" property is not valid for

convergent series: i.e., replacement of each term by the sum (or difference)

of two other terms may alter the character of the series.

Theorem 4.10 If each term of a convergent series is multiplied by a

constant k, the resulting series is also convergent and its sum is that of the

original series multiplied by k. If the first series diverges and k f 0, the

new series also diverges (distributive property for series).

Proof Let L~ an be the first series and L~ kan the second. If An and A~

are the corresponding partial sums, we have A~= kAn. Hence if An --t A,

it follows that A~ --t kA. If An --t oo and k f 0, then A~ --t oo. If An

does not tend to a limit (finite or infinite), then A~ does not tend to a

limit either, in view of An = k-^1 A~.

Theorem 4.11 If a finite number of terms with sum Sare omitted from

a series, or added to the series, the, character of the series does not change.

If the original series is convergent with sum A, the new series is also

convergent with sum A - S or A+ S, respectively.

Proof The conclusion follows from A~ =An =i= S, where n is large enough

so that An will contain all the omitted terms, or A~ all the added terms.

-,

Theorem 4.12 Any linear combination of the terms of two convergent

series :Z::~ an and :Z::~ bn, with constant coefficients k, k', yields a series

L~(kan + k'bn) which is also convergent and has kA + k' B as a sum,

where A and B are the sums of the two given series. In particular, adding

or subtracting term by term the two series results in a series converging

to A + B or A - B, respectively.

Proof As in the real case, it follows at once by taking limits as n --t oo
in the identity

n

Sn= L(kaj + k'bj) = kAn + k'Bn

j=l

Theorem 4.13 If A = :Z::~ an and B = L~ bn are both absolutely

convergent, then every series Lhk=l ajbk containing just once the product
of each term aj by each term bk converges absolutely and has sum equal
to AB.

Proof Suppose that the series :Z::j,'k=l ajbk is formed in the following
manner:

( 4.8-2)