1549901369-Elements_of_Real_Analysis__Denlinger_

8.7 Analytic Functions 531

(b) \In E .N, jtl Cj = jtl c~ aij)

= i~ Ct

1

aij) (by the linearity of series, Theorem 8.1.12)

~ f (f aij) = f Ri.

i=l j=l i=l

oo n oo

Therefore, L Cj converges and L Cj ~ L Ri.

j=l j=l i=l

n n(oo) oo ( n ) oo (oo ) oo

Similarly, i~ Ri = i~ j~l aij = j~l i~ aij ~ j~l i~ aij = j~l Cj·

*Theorem 8.7.17 (Absolute Convergence of Double Series) Suppose
00
the sum by rows of the double series L Jaij I converges. Then both the sum
i,j=l
00
by rows and the sum by columns of the double series L aij converge and are
i,j=l
n oo
equal. In fact, L ~ and L cj converge absolutely.
i=l j=l
00
Proof. Suppose the sum by rows of the double series L laij I converges.
i,j=l
Using the notation of Definition 8.3.8, let atj = max{ aij, O} and a;j =
max{-aij, O}. By Lemma 8.3.9 and Theorem 8.3.10,

00

(Both of these nonnegative series converge by comparison with L Jaij J.)

i,j=l

Consider the two matrices

af1 a12 a13 + +. .. alj + a!1 a!2 a!3 a!j

ai1 a22+ a 23 +... aij a21 a22 a; ... a;-j

a31 a32 + + a33 + ... ajj a31 a;2 a33 ... a3j

and

ail + ai2 + ai3 + aij + ail. ai2 aiJ a;j

(36) (37)