10.5 Double Series
Sometimes it is required to consider double series which are of the form
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
In other words, when does it make no difference which subscript is summed over first? In
the case of finite sums there is no issue here. You can always write
because addition is commutative. However, there are limits involved with infinite sums
and the 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 of thumb is this: If something involves changing the order
in which two limits are taken, you may not do it without agonizing over the
question. In general, limits foul up algebra and also introduce things which are
counter intuitive. Here is an example. This example is a little technical. It is
placed here just to prove conclusively there is a question which needs to be
Example 10.5.1 Consider the following picture which depicts some of the ordered
where m,n are positive integers.
The numbers next to the point are the values of amn. You see ann
= 0 for all n,
= c for
on the line y
= 1 + x whenever m >
= −c for all
on the line y
= x −
1 whenever m >
m=1∞amn = a if n = 1, ∑
m=1∞amn = b − c if n = 2 and if
n > 2,∑
m=1∞amn = 0. Therefore,
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,
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
Don’t become upset by this. It happens because, as indicated above, limits are taken
in two different orders. An infinite sum always involves a limit and this illustrates why
you must always remember this. This example in no way violates the commutative law of
addition which has nothing to do with limits. However, it turns out that if aij ≥ 0
for all i,j, then you can always interchange the order of summation. This is
shown next and is based on the following lemma. First, some notation should be
Definition 10.5.2 Let f
for a ∈ A and b ∈ B where
A,B are sets which means that f
is either a number, ∞, or −∞. The symbol,
+∞ is interpreted as a point out at the end of the number line which is larger
than every real number. Of course there is no such number. That is why it is called
∞. The symbol, −∞ is interpreted similarly. Then
where Sb ≡
Unlike limits, you can take the sup in different orders.
Lemma 10.5.3 Let f
for a ∈ A and b ∈ B where A,B are sets.
Proof: Note that for all a,b, f
and therefore, for all
Repeat the same argument interchanging a and b, to get the conclusion of the lemma.
Theorem 10.5.4 Let aij ≥ 0. Then
Proof: First note there is no trouble in defining these sums because the aij are all
nonnegative. If a sum diverges, it only diverges to ∞ and so ∞ is the value of the sum.
Next note that
because for all j,
Therefore, using Lemma 10.1.3,
Interchanging the i
in the above argument proves the theorem. ■
The following is the fundamental result on double sums.
Theorem 10.5.5 Let aij ∈ ℝ and suppose
and every infinite sum encountered in the above equation converges.
Proof:By Theorem 10.5.4
Therefore, for each j,∑
and for each i,∑
on Page 651
both converge, the first one for every j
the second for every i.
so by Theorem 10.2.2 again,
both exist. It only remains to verify they are equal.
By Theorem 10.5.4 and Theorem 10.1.6 on Page 646
It follows the two series are equal. ■
One of the most important applications of this theorem is to the problem of
multiplication of series.
Definition 10.5.6 Let ∑
i=r∞ai and ∑
i=r∞bi be two series. For n ≥ r,
The series ∑
n=r∞cn is called the Cauchy product of the two series.
It isn’t hard to see where this comes from. Formally write the following in the case
r = 0:
and start multiplying in the usual way. This yields
and you see the expressions in parentheses above are just the cn for n = 0,1,2,
Therefore, it is reasonable to conjecture that
and of course there would be no problem with this in the case of finite sums but in the
case of infinite sums, it is necessary to prove a theorem. The following is a special case of
Theorem 10.5.7 Suppose ∑
i=r∞ai and ∑
j=r∞bj both converge
Proof: Let pnk = 1 if r ≤ k ≤ n and pnk = 0 if k > n. Then
Therefore, by Theorem 10.5.5