1549901369-Elements_of_Real_Analysis__Denlinger_

8.1 Basic Concepts and Examples 457

Theorem 8.1.10 (Grouping by Inserting Parentheses)

(a) If'L, an converges, then any grouping L, bn formed from L, an by inserting

parentheses also converges, and has the same sum.

(b) If L, an has a grouping that diverges, then L, an diverges.

(c) Some divergent series can be grouped by inserting parentheses to f orm a

convergent series.

Proof. Exercise 15. •

Actually, we have already seen an application of this theorem in Chapter

2. Our proof that the h armonic series diverges, given in Section 2.5, used the
   method of inserting parentheses.
   The remaining two theorems of this section are adaptations of theorems
   about sequences to the context of series. The first of these theorems is a straight-
   forward adaptation of the Cauchy criterion of sequences.

Theorem 8.1.11 (Cauchy Criterion for Convergence of Series) Ase-

ries fan converges iff Ve> 0, :lno EN 3 n > m ~no::::} I f: akl < E:.

n=l k=m+I

Proof. Exercise 16. •

The next theorem is a straightforward adaptation of the algebra of limits
of sequences to the context of series.

00 00
Theorem 8.1.12 (Linearity of Sums of Series) If L, an and L, bn
n=l n=l
00 00
converge, and c E JR, then both L, (an+ bn) and L, can converge, and
n=l n=l

00 00 00

(a) L (an+ bn) = L an+ L bn

n=l n=l n=l

00 00

(b) L can = c L an.

n=l n=l

Proof. Exercise 17. •