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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