We need general techniques for determining
when a series converges.
[MUSIC].
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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