WEBVTT
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We now continue our discussion
of infinite series.
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Sometimes we have to deal with
series where the terms being
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added are indexed by multiple
indices, as in
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this example here.
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We're given numbers, aij,
and i ranges over all
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the positive integers.
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j also ranges over all the
positive integers.
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So what does this
sum represent?
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We can think of it as follows.
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We have here a two-dimensional
grid that corresponds to all
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the pairs (i,j).
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And in essence, each one of
those points corresponds to
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one of the terms that
we want to add.
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So we can sum the different
terms in some arbitrary order.
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Let's say we start from here.
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Take that term, add this term,
then add this term here, then
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add this term, then the next
term, next term, and so on.
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And we can keep going that
way, adding the different
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terms according to
some sequence.
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So essentially, what we're doing
here is we're taking
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this two-dimensional grid and
arranging the terms associated
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with that grid, in some
particular linear order.
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And we're summing those
terms in sequence.
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As long as this sum converges to
something as we keep adding
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more and more terms, then
this double series
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will be well defined.
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Notice, however, we can
add those terms in
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many different orders.
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And in principle, those
different orders might give us
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different kinds of results.
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On the other hand, as long as
the sum of the absolute values
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of all the terms turns out to be
finite, then the particular
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order in which we're adding the
different terms will turn
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out that it doesn't matter.
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There's another way that we can
add the terms together,
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and this is the following.
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Let us consider fixing a
particular choice of i, and
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adding all of the terms that
are associated with this
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particular choice of i, as j
ranges from 1 to infinity.
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So what we're doing is we're
taking the summation from j
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equal to 1 to infinity, while
keeping the value of i fixed.
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We do this for every
possible i.
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So for every possible i,
we're going to get
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a particular number.
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And then we take the numbers
that we obtain for the
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different choices if i, so i
ranges from 1 to infinity.
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And we add all those
terms together.
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So this is one particular order,
one particular way of
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doing the infinite summation.
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Now, why start with the
summation over j's while
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keeping i fixed?
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There's no reason for that.
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We could also carry out the
summation by fixing a
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particular choice of j and
summing over all i's.
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So now it is i that ranges
from 1 to infinity.
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And we look at this
infinite sum.
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This is the infinite
sum of those terms.
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We obtain one such infinite
sum for every choice of j.
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And then we take that sum that
we obtain for any particular
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choice of j, and add
over the different
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possible values of j.
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So j goes from 1 to infinity.
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This is a different way of
carrying out the summation.
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And these are going to give us
the same result, and the same
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result that we would also obtain
if we were to add the
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terms in this particular order,
as long as the double
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series is well-defined, in
the following sense.
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If we have a guarantee that the
sum of the absolute values
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of those numbers is finite, no
matter which way we add them,
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then it turns out that we can
use any particular order to
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add the terms in the series.
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We're going to get
the same result.
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And we can also carry out the
double summation by doing--
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by adding over one
index at a time.
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A word of caution--
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this condition is not
always satisfied.
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And in those cases, strange
things can happen.
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Suppose that the sequences we're
dealing with, the aij's,
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take those particular values
indicated in this picture.
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And all the remaining terms, the
aij's associated with the
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other dots, are all 0's.
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So all these terms out
there will be 0's.
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If we carry out the summation by
fixing a j and adding over
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all i's, what we get here is 0,
and a 0, and a 0, and a 0.
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That's because in each row we
have a 1 and a minus 1, which
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cancel out and give us 0's.
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So if we carry out the summation
in this manner, we
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get a sum of 0's, which is 0.
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But if we carry out the
summation in this order, fix
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an i, and then add over all j's,
the first term that we
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get here is going to be 1,
because in this column, this
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is the only non-zero number.
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And then in the remaining
columns, as we add the terms,
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we're going to get 0's,
and 0's, and so on.
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And so if we carry out the
summation in this way, we
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obtain a 1.
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So this is an example that shows
you that the order of
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summation actually may matter.
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In this example, the sum of the
absolute values of all of
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the terms that are involved is
infinity, because we have
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infinitely many plus or minus
1's, so this condition here is
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not satisfied in this example.
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Let us now consider the case
where we want to add the terms
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of a double sequence, but over
a limited range of indices as
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in this example, where we have
coefficients aij, which we
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want to add, but only for those
i's and j's for which j
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is less than or equal to i.
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Graphically, this means that we
only want to consider the
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pairs shown in this picture.
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So these points here correspond
to i,j pairs for
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which i is equal to j.
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Terms on the right, or points
to the right, correspond to
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i,j pairs for which i is
at least as large as j.
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We can carry out this summation
in two ways.
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One way is the following.
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We fix a value of i, and
we consider all of the
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corresponding terms, that
correspond to different
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choices of j.
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But we only go up to the point
where i is equal to j.
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This is the largest term.
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So what are we doing here?
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We're taking the coefficients
aij, and we are adding over
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all j's, starting from 1, which
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corresponds to this term.
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And j goes up to the point where
it becomes equal to i.
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We do this for every
value of i.
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And so we get a number for the
sum of each one of the
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columns, and then we add
those numbers together.
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So we're adding over all
i's, and i ranges
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from 1 up to infinity.
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This is one way of carrying
out the summation.
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Alternatively, we could fix a
value of j, and consider doing
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the summation over
all choices of i.
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So this corresponds to
the sum over all
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choices of i, from where?
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The smallest term, the first
term, happens when i is equal
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to the value of j.
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And then we have larger choices
of i, numbers for
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which i is bigger than the
corresponding value of j.
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And so i ranges from j all
the way to infinity.
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And this is the sum over one of
the rows in this diagram.
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We do this for every j.
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We get a result, and then
we need to add all
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those results together.
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So we're summing for all j's
from 1 up to infinity.
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So these are two different ways
that we can evaluate this
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series associated with
a double sequence.
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We can either add over all j's
first and then over i's, or we
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can sum over all i's first,
and then over all j's.
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The two ways of approaching this
problem, this summation,
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should give us the
same answer.
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And this is going to be, again,
subject to the usual
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qualification.
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As long as the sum of the
absolute values of the terms
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that we're trying to add
is less than infinity--
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if this condition is true, then
the two ways of carrying
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out the summation are equal, and
they're just two different
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alternatives.