Coshi condensation.
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Let's think back to the harmonics series.
Well here's the harmonic series.
And we proved that it diverges.
And how do we do that?
Well, we did this grouping trick.
Here's the beginning of the harmonic
series and it keeps on going.
What we saw was that we got a one here and
a half here.
This third and this fourth can be grouped
together and the next four
terms can be grouped together.
And then the next eight terms will group
together and so on.
But this this group here is at least a
half.
because it's two terms that are each as
big as a fourth.
This next group here is also at least a
half,
because it's four terms that are each at
least an eighth.
And then the next group of eight terms is
at least a half, and the next group
of 16 terms is at least a half.
So this is even worse than adding up 1
plus a half
plus a half plus a half plus a half, and
that series diverges.
So we're able to show that the harmonic
surge diverges by
doing this grouping into piles with sizes
the power of two.
This isn't just some one off trick, this
is
a technique that we can generalize and
then apply broadly.
That we used it for the
harmonic series to show that the series
diverged.
But we could also use the same kind of
thinking to show that a series converges.
The name for this trick is Cauchy
Condensation.
Here's how it goes.
At least when using it to prove
convergence.
Let's suppose that I've got a decreasing
sequence
and that all of the terms are positive.
I want to know does this series, the sum k
goes from one to
infinity of a sub k converge or diverge?
I can group them by powers of two.
So what I mean is I'll put a sub one by
itself.
I'll group together a-sub-2 and a-sub-3.
I'll group together a-sub-4, a-sub-5,
a-sub-6, and a-sub-7.
I'll group a-sub-8 alongside a-sub-9,
a-sub-10, a-sub-11, a-sub-12, a-sub-13,
a-sub-14,
a-sub-15.
And then I'll group a sub 16, a sub 17.
I mean, I keep on going until I get a sub
31 and then I start a new group at a sub
32.
Now I'll overestimate the groups.
Well, how big is a sub 2 plus a sub 3?
The key fact here, is that the sequence of
the a sub k's is decreasing.
That means that a sub 3 is smaller than a
sub 2.
So, if I add a
sub 2 and a sub 3, the result is less than
twice a sub 2.
What about this next group.
What's a sub 4 plus a sub 5 plus a sub 6
plus a sub 7?
Well, a sub 5, a sub 6 and a sub 7 are all
less than a sub 4.
Again, because the sequence is decreasing.
So, if I add these 4 numbers up, the
result
is less than if I just multiplied the
fourth term
by 4.
What about the next eight terms?
Well, these terms, from a sub 9 through a
sub 15, are all smaller than a sub 8.
So these eight terms, all together are
less, than just
copies of 8 sub 8.eight and it keeps on
going, right.
The next 16 terms, if I were to group them
all together, would be less than
16 copies of a sub 16 and so on.
Let me write down a precise statement of
what we're doing.
So here's how I can write down the
grouping in symbols.
Look at the sum of the a sub k's, k goes
from
1 all the way up to 1 less a power of 2.
So 2 to the n minus 1 say.
So this would be like adding up the first
seven
terms, or the first 15 terms, or the first
31 terms.
It's a collection of terms where I can
actually do the grouping.
Well, this is less than or equal to, then,
the
sum of what happens after I do the
grouping, right?
Which is 2 to the k times a sub 2 to the
k.
This is what I get when I do this
grouping.
K goes from 0, all the way up to n minus
1.
Why is this significant?
So, we were originally
interested in studying this series and
determining whether this series converged
or diverged.
And now, we're led to studying this
series.
The sum k goes from 0 to infinity of 2 to
the k times the 2 to the kth term in the
sequence.
I'm going to call this series the
condensed series associated to this
series.
Now what if the condensed series
converges?
So,
if this condensed series converges right,
then what happens?
Recall what I know.
This condensed series is over estimating
the original series.
So, if I'm trying to study something about
the
partial sums of the original series, what
I know
now is that those partial sums for the
original
series are bounded above by the value of
the
convergence condense series.
I also know that the partial sums for the
original series are
increasing and that's just because all of
the a sub k's are positive.
So as I add up more and more of the
a sub k's, I'm getting a bigger and bigger
number.
So the sequence of partial sums is
increasing.
That means the sequence of partial sums
is bounded above and increasing and
therefore convergent.
And what that means, then is
that the original series that I'm
interested in converges.
Let's summarize the theorem.
So here's a statement of the Cauchy
Condensation test.
You've got a sequence, which is decreasing
and all the terms are positive.
Then the series, sum k goes from one to
infinity a sub k converges,
if the condensed series converges.
That's what we just proved.
And in
fact, it's an if and only if.
So the fate of the condensed series is
exactly the same as the fate of the
original series.
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