We need general techniques for determining when a series converges.
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We've already seen an example of what's usually called the Comparison Test.
The original series that we looked at was this series.
It has the sum, n goes from 0 to infinity.
Sine squared n, divided by 2 to the n. And that series converges.
But, remember why.
Let's recall how we prove that this series converges.
And what we did was to compare it to a geometric series.
So, 0 is less than or equal to sine
squared n over 2 to the n, is less than or equal to 1 over 2 to the n.
And these inner qualities just follow from the fact
that sine squared is trapped between 0 and 1.
And what do I know about the sum of 1 over 2 to the n?
The sum 1 over 2 to the n, n goes from 0 to infinity, converges.
Right, this is a geometric series, I can even evaluate it.
Now, what does
that tell me about the original series that I'm studying?
What about the sine squared n over 2 to the n?
Well the sequence or partial sums for this series are
all bounded above by the vast value of this series.
And the sequence of partial sums is non
decreasing because these terms are all none negative.
So, I've got a monotone sequence which is bounded above.
That means the sequence of partial sums converges, and that's just what it means
to say that this series converges. That was a bit quick.
So let's generalize this argument, try to
formulate it as a broadly useful technique.
So here's the usual setup.
I'm, imagine that I've got two series. I'll use K as the index, K goes from 0 to
infinity of a sub k, and another series, K goes from 0 to infinity of b sub k.
And I'm supposing that 0 is less than or equal
to a sub k, is less than or equal to b sub k, for all K.
I'm imagining that the sum of the b sub k's converges.
Right.
So I'm going to be assuming that this series,
converges, and then I'm wondering about this series.
Does that series converge, or diverge? Well, to study that question, supposed to
look at the sequence of partial sums. So that's S sub n is the sum
of the terms, K goes from 0 to n, of a sub k.
And now, the whole question is whether this sequence converges?
Because the convergence of this sequence is exactly
what it means to say that this series converges.
What kind of sequence is that? Well, here's what I know.
I know that a sub k is less than or equal to b sub k.
And that means if I add up a whole bunch of a sub k's.
Say, the a sub k is between k equals 0 and n.
That's less than or equal to adding up the b sub k's, K goes from 0 to n.
Now this thing here is just the nth partial sum.
Now what do I know about this sum?
Well the rest of the b sub k's are all non negative, so if I add up even more b sub
k's, I'm just making this even bigger. So what I can conclude,
is this. That s sub n is less than or
equal to the value of the series, K goes from 0 to
infinity, of b sub k. In other words, what I can conclude is
that S sub n, is bounded above. And it's monotone.
So let's check that.
I'm going to show that if m is bigger than n, then
S sub m is at least as big as S sub n.
Well, what's S sub m?
S sub m is the sum, K goes from zero to m, of a sub k.
And I'm sure that's bigger than or equal to the sum K goes from 0 to n, a sub k.
But, m is bigger than n,
so I can re-write this sum. I can split this sum up.
This is the sum K goes from 0 to n, of
a sub k, plus the terms between n plus 1 and m.
So, K goes from n plus 1 to m of a sub k, and that's at least as big, I hope,
as the sum K goes from 0 to n of a sub k. Now, why is this true?
Well,
I've got the sum K goes from 0 to n of a
sub k on both sides, but I've got this extra term over here.
What I need to know is that this extra term is at least 0.
But that's true, because all the a sub k's are non negative, and if you
add up a whole bunch of non negative
numbers, well, then, the result is non negative.
So this is telling me that the sequence of partial sums, is non decreasing.
So the
sequences of partial sums is non decreasing and bounded above.
So with a monotone convergence theorem, it converges.
Let's summarize that.
So here's what we've proved. If we've got two sequences, which are
related in this way, that a sub k is between 0 and b sub k, and the sum, K goes
from 0 to infinity, of b sub k, converges. Then,
the sum K goes from 0 to infinity of a sub k, also converges.
We can also say something when the sum of the a sub k's, diverges.
Suppose that 0 is less than or equal to a sub k, is less than or equal to b sub k.
And let's suppose that the sum of the a
sub k's, K goes from 0 to infinity, diverges.
And I don't know, what can I say about the sum of the b sub k's?
Well, if this series diverges, that means that the sequence of partial sums,
the sum, K goes from 0 to n of a sub k. That can't converge,
right, because to say this thing converges is to say the
series converges, so if this series diverges, this sequence must diverge.
But that sequence, is monotone for the same reason as before, right.
I'm adding up non negative terms, so as I
add up more terms, at least is non decreasing.
So the sequence is a monotone sequence, and yet, that sequence doesn't converge.
So that means that the sequence of partial sums must be
unbounded, because if it were bounded, and it was
monotone, then it would converge, but it doesn't converge.
So I've got an unbounded sequence.
What about the sequence of partial sums for
the series involving the b sub k's, right?
So the sequence of partial sums there as the sum K goes from 0 to n of b sub k.
This is even bigger than the sequence
of partial sums for the a sub k's. This sequence, is likewise unbounded.
Well, if this sequence is unbounded then that sequence doesn't converge.
And if that sequence doesn't converge that means
that the sum of the b sub k's, diverges.
And that was fast.
So we hope to see the convergence and divergence statements together.
Well, here's our standing assumption. That 0 is less than or
equal to a sub k, is less than or equal to b sub k.
And the first thing that we saw, is that if the series of
the b sub k's converges, then the series of the a sub k's converges.
On the other hand, if the series of the a sub
k's diverges, then the series of the b sub k's diverges.
It's super important to keep track of the directions of these implications.
Alright, here I've got the convergence
of b sub k implying the convergence of a sub k.
And here I've got the divergence of a sub k implying the divergence of b sub k.
And hopefully that makes sense, right?
This is saying that as long as you've got non
negative terms, if you're below a convergent series, you converge.
And this is saying, as long as you've got non
negative terms, if you're above a divergent series, you also diverge.
And let me end by saying
the Comparison Test is just a ton of fun to use.
It's definitely my favorite test.
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