5.4 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 when
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 5.4.1 Consider the following picture which depicts some of the ordered pairs
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 >
1, and amn
= −c for
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.
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 desired.
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 5.4.2 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
Unlike limits, you can take the sup in different orders.
Lemma 5.4.3 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.
Lemma 5.4.4 If
is an increasing sequence in
Proof: Let sup
In the first case, suppose r < ∞.
Then letting ε >
be given, there exists n
such that An ∈
is increasing, it follows if
m > n,
then r −ε < An ≤ Am ≤ r
and so limn→∞An
as claimed. In the case where r
is a real number, there exists n
such that An > a.
is increasing, it follows that
m > n, Am > a.
But this is what is meant by limn→∞An
The other case
is that r
But in this case, An
for all n
and so limn→∞An
Theorem 5.4.5 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
because for all j,
Interchanging the i
in the above argument proves the theorem. ■
The following is the fundamental result on double sums.
Theorem 5.4.6 Let aij ∈ F and suppose
and every infinite sum encountered in the above equation converges.
Proof:By Theorem 5.4.5
Therefore, for each j,∑
and for each i,∑
By Theorem 5.1.9
on Page 202
both converge, the first one for every j
and the second
for every i.
so by Theorem 5.1.9 again,
both exist. It only remains to verify they are equal. By similar reasoning you can replace aij
with Reaij or with Imaij in the above and the two sums will exist.
The real part of a finite sum of complex numbers equals the sum of the real parts. Then
passing to a limit, it follows
Note 0 ≤
Therefore, by Theorem 5.4.5
and Theorem 5.1.5
Similar reasoning applies to the imaginary parts. Since the real and imaginary parts of the
two series are equal, it follows the two series are equal. ■
One of the most important applications of this theorem is to the problem of multiplication
Definition 5.4.7 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,
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 Merten’s
Theorem 5.4.8 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 5.4.6