94 CHAPTER 5. INFINITE SERIES OF NUMBERS

In other words, first sum on j yielding something which depends on k and then sum these.The major consideration for these double series is the question of when

∞

∑k=m

∞

∑j=m

a jk =∞

∑j=m

∞

∑k=m

a jk.

In other words, when does it make no difference which subscript is summed over first? Inthe case of finite sums there is no issue here. You can always write

M

∑k=m

N

∑j=m

a jk =N

∑j=m

M

∑k=m

a jk

because addition is commutative. However, there are limits involved with infinite sums andthe interchange in order of summation involves taking limits in a different order. Therefore,it is not always true that it is permissible to interchange the two sums. A general rule ofthumb is this: If something involves changing the order in which two limits are taken, youmay not do it without agonizing over the question. In general, limits foul up algebra andalso introduce things which are counter intuitive. Here is an example. This example is alittle technical. It is placed here just to prove conclusively there is a question which needsto be considered.

Example 5.5.1 Consider the following picture which depicts some of the ordered pairs(m,n) where m,n are positive integers.

...0 0 c 0 −c0 c 0 −c 0b 0 −c 0 00 a 0 0 0

· · ·

The a,b,c are the values of amn. Thus ann = 0 for all n≥ 1, a21 = a,a12 = b,am(m+1) =−cwhenever m > 1, and am(m−1) = c whenever m > 2. The numbers next to the point are thevalues of amn. You see ann = 0 for all n, a21 = a,a12 = b,amn = c for (m,n) on the liney = 1+ x whenever m > 1, and amn = −c for all (m,n) on the line y = x− 1 wheneverm > 2.

Then ∑∞m=1 amn = a if n = 1, ∑

∞m=1 amn = b− c if n = 2 and if n > 2,∑∞

m=1 amn = 0.Therefore, ∑

∞n=1 ∑

∞m=1 amn = a+b−c. Next observe that ∑

∞n=1 amn = b if m= 1,∑∞

n=1 amn =a+ c if m = 2, and ∑

∞n=1 amn = 0 if m > 2. Therefore, ∑

∞m=1 ∑

∞n=1 amn = b+ a+ c and so

the two sums are different. Moreover, you can see that by assigning different values of a,b,and c, you can get an example for any two different numbers desired.

Don’t become upset by this. It happens because, as indicated above, limits are takenin two different orders. An infinite sum always involves a limit and this illustrates whyyou must always remember this. This example in no way violates the commutative law ofaddition which has nothing to do with limits. Algebra is not analysis. Crazy things happenwhen you take limits. Intuition is routinely rendered useless.

However, it turns out that if ai j ≥ 0 for all i, j, then you can always interchange theorder of summation. This is shown next and is based on the Lemma 2.10.5 which says youcan intercange supremums.