530 Chapter 8 • Infinite Series of Real Numbers
00
is called the sum by rows of L aij. If all the column sums (series) Cj
i ,j=l
converge, then the series
00
is called the sum by columns of L aij·
i,j=l
Example 8.7. 15 To see that the sum by rows and the sum by columns can
be quite different, consider the matrix
1 1 1 1 10 -~ -2^1 -2^1 -2^1
(^0 0) - 4^1 -4^1 -4^1
(^0 0 0) -3^1 -3^1
Here, the sum by columns is 2 but the sum by rows diverges. (Exercise 17.)
- Lemma 8. 7. 16 Suppose all the entries of (35) are nonnegative. If every row
00
sum Ri converges and the "sum by rows" LR converges, then
i=l
(a) every column sum Ci converges, and
(b) the sum by columns converges and equals the sum by rows; i.e.,
00 00
2:=~= l:=Ci,
i=l j=l
or f: ( f: aij) = f: (f: aij).
i=l j=l j=l i=l
Proof. Suppose all the entries of (35) are nonnegative, every row sum Ri
00
converges, and the "sum by rows" L Ri converges.
i=l
00
(a) Let i,j E N. Then since the terms are all nonnegative, aij < L aij,
j=l
00
which converges (to Ri), so aij :::; Ri· Therefore, Vj E N, L aij converges by
i=l
00
comparison with the series L Ri. That is, every column sum converges.
i=l