494 Chapter 8 • Infinite Series of Real NumbersBy Theorem 8.4.4, L ck must converge. Prove that this convergence is
not absolute by showing that Yk, lckl 2 t·- State and prove the commutative law for the Cauchy product of series.
- State and prove the distributive law for sums and Cauchy products of
series. - State and prove the associative law for the Cauchy product of series.
- Prove the claims made in (17) and (18) of the proof of Theorem 8.4.4.
8.5 Series of Products
In the previous section we studied products of series; in this section we study
series of products. In particular, we study series of the form L ck = L akbk
and investigate conditions on the sequences { ak} and {bk} that will guarantee
convergence of such a series.
To determine whether L akbk converges, it is not enough to check whether
both L ak and L bk converge. Indeed, it is easy to find convergent series L ak
and .Ebk for which .Eakbk diverges (see Exercise 1). It is also easy to find
divergent series E ak and L bk for which L akbk converges (Exercise 2). How-
ever, with the help of the following theorem, we can easily prove that if L ak
and L bk are both absolutely convergent, then so is L akbk (Exercise 4).
Theorem 8.5.1 If L ak converges absolutely and {bn} is a bounded sequence,
then L akbk converges absolutely.
Proof. Exercise 3. •We seek weaker conditions on the sequences { ak} and {bk} that will guar-
antee convergence of the series L akbk. The following result will prove very
useful in that investigation.
Theorem 8.5.2 (Abel's Summation by Parts Formula) Let {ak} and
{bk} be sequences, and define
n
So= 0, and "in?: 1 Sn= L ak.
k=l
Then, for all l :<:::: m < n,
n n
L akbk = L Sk(bk - bk+d + Snbn+I - Sm-lbm.
k=m k=m
Proof. Let {ak}, {bk}, and {Sn} satisfy the hypotheses. Then, for all k?: 0,
akbk = (Sk - Sk-1)bk
= Sk(bk - bk+1) + Skbk+1 - Sk-1bk.