8.4 The Cauchy Product of Series 487terms, subtracting the next two terms, and so on. How does this differ
from the series in Exercise 14? Prove that the resulting series converges.- Beginning with the harmonic series, form a new series by adding the first
two terms, then subtracting the next term, then adding the next two
terms, subtracting the next term, and so on-always adding two terms
and subtracting the next term. Prove that the resulting series diverges. - By Theorem 8.3.14 (a), the alternating harmonic series can be rearranged
to a series that diverges to +oo. Write out the first 24 terms of the rear-
ranged series described in the proof of this theorem. [Use a calculator or
computer to check the required inequalities.] - By Theorem 8.3. 14 (b), the alternating harmonic series can be rearranged
to a series that converges to 0. Write out the first 24 terms of the rear-
ranged series described in the proof of this theorem. [Use a calculator or
computer to check the required inequalities.] - Prove that although the alternating series f: ( -1 )n+l n +
1
diverges, the
n=l n
sequence obtained by grouping consecutive terms in pairs (using paren-
theses) converges absolutely. [See Theorem 8.1.10 (c).]8.4 The Cauchy Product of Series
Given convergent series L ak and L bk, it is often desirable to express their
product as another convergent series:
We seek an appropriate definition of such a product series I: Ck, a formula
for its terms Ck, and conditions under which the product series converges. To
get a good start in that direction we take a look at a simpler, finite case:
Because multiplication of one sum by another must obey the distributive
law, each term in the one sum must be multiplied by each term in the other
sum, and then all these products must be added together. Thus, (a1 + a2 +
a3)(b 1 + b 2 + b3 + b 4 ) is the sum of all twelve entries in the following matrix:
[)lr[0~--~1-lf~_,~1-lfi_,~l-b-4:]
,,.arits1 ,,arifh _g,.~-lf 3 g:Ar4.
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