Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this fifth part--part five of five--we cover a calculus for sequences, numerical methods, series and convergence tests, power and Taylor series, and conclude the course with a final exam. Learners in this course can earn a certificate in the series by signing up for Coursera's verified certificate program and passing the series' final exam.
Infinite Series
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來自 University of Pennsylvania 的課程
Calculus: Single Variable Part 5 - Discrete Calculus
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從本節課中
Series and Convergence Tests
In "ordinary" calculus, we have seen the importance (and challenge!) of improper integrals over unbounded domains. Within discrete calculus, this converts to the problem of infinite sums, or <b>series</b>. The determination of convergence for such will occupy our attention for this module. I hope you haven't forgotten your big-O notation --- you are going to need it!
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Robert GhristProfessor
Mathematics and Electrical & Systems Engineering
Welcome to calculus. I'm professor Ghrist.
We're about to begin lecture 50 on infinite series.
The next few lessons are at the heart of this chapter
and deal with infinite series, a hopefully not too unfamiliar topic.
We're going to detail specific examples, and consider
convergence and divergence of series, in the same
manner that we dealt with convergence and divergence of improper integrals.
We continue with our program of building a calculus for sequences.
Considering, in this lesson, the analog of the improper integral.
These are called infinite series. Something that you've seen before.
But now, we're going to treat them a bit more carefully.
Now, how was it that we handled improper integrals?
Recall, the integral from 1 to infinity of f of x dx
was defined in terms of a limit. The limit as t goes to infinity
of the finite integral of f from 1 to
t. So, in defining what we mean by
an infinite series, that is, the sum as n goes from 1 to infinity of a sub n.
We'll use the same approach.
We can think of this infinite series of being something like a
discretization of an improper integral. And so, we can
define this sum to be the limit as t goes to infinity of
the sum n goes from 1 to t of a sub n.
That is, the series is really the limit of the sequence of partial sums.
Recall that, when it comes to improper
integrals, the central and
subtle question is that of convergence or divergence.
The same occurs with infinite series.
Now, you've seen infinite series all
throughout this course, from the very beginning.
And I have told you repeatedly, don't worry too much about convergence
or divergence, we'll worry about that at the end of the course.
Well, it is time to push that button. It is time for all of those
worries that have been inside your head to come out, and we will deal with them.
We will go through the notion
of convergence or divergence, slowly and carefully.
Giving you lots of practice, so that you get good at it.
Let's consider some examples of series. Some are very easy to understand.
If we look at 1 plus 1 over e plus 1 over e squared plus 1 over e cubed, et cetera.
Well, we recognize this as a geometric series.
So, not only do we know that it converges, we know exactly to what value.
On the other hand, we recall that the geometric series does not always converge.
For example,
if we evaluate the geometric series at negative 1, then,
one could try to argue that the appropriate value is
0, or, one could argue that the appropriate convergence
is to 1, or even to 1 half. None of these
holds. This is a divergent series, because the
limit of partial sums does not exist. It oscillates.
Now, subseries do converge, but are so subtle in how they do so.
They're very difficult to evaluate. The sum from 1 to
infinity of 1 over n squared does converge as we'll be able to show soon.
But it convereges to
a value that is very difficult to determine
exactly. And that value is pi squared over 6.
Some series are not so easy to figure out. Consider 1 plus
1 half plus 1 3rd plus 1 4th plus 1
5th, et cetera. To what, if anything, does this converge.
This last example is a special series that is called the harmonic series.
Let's investigate this series a little bit.
We'll fire up the computer and see what happens
when we add, let's say the first 10,000 terms together.
We get a result that seems decidedly finite.
And it's about 9.79. But what happens
if we say add another 5,000 terms to that. Well,
our result is only a little bit bigger. We would
guess that this is about to converge, since the next
term in the series is 1 over 15,001.
That is quite small. But let's just type it.
Let's add the first 20,000 terms together.
Well, we can see that we're slowing down a bit.
We've only gone about 0.3 more, not even that.
Still I'm not going to be quite comfortable
until I see that it's converged to several
decimal places.
But this does not appear to be happening even with 25,000 terms.
So let's sum up the first 100,000 thousand terms.
Surely, surely, if this diverges and goes off to infinity, then we would see it.
Well, the answer in this case is just a little bit more than we had before,
but not so little as to make me conclude that it converges.
So what is happening in this case?
Well, we're going to have to use something beyond
a calculator to determine convergence or divergence in this case.
What we're going to use is our calculus intuition.
Harmonic series reminds
us of the improper integral, as x goes from 1
to infinity of 1 over x dx. Indeed
we can consider that sum at series as a
discretization of this integral.
As a left Riemann sum of this integral with a uniform
partition of step size one.
Now, we know that the integral of 1 over x as x goes from 1 to infinity diverges.
And we can conclude from the geometry of this diagram if nothing else, that
the harmonic series is strictly bigger. Therefore, we
can conclude that harmonic series diverges.
What's wonderful about this is that we viewed our continuous or
smooth calculus understanding to conclude things about this
discreet calculus. Example, and this discussion raise
the question, what can you trust when it comes to determining
convergence or divergence? As we've seen, a calculator is not
the best tool for determining convergence or divergence.
What about your intuition?
Maybe you should just get a feel for what works.
Well, intuition is also flawed. Let's look at an example.
I claim that the series 1 minus a half plus
a third minus a fourth, plus a fifth, et cetera,
something that we'll call the alternating harmonic series, does converge, and it
converges to a value of log of 2. Now, why is that the case?
Well, recall that log of 1 plus x is exactly this kind
of alternating sum of x to the n over n. If we
evaluate this at x equals 1, then we obtain this value.
Now, technically speaking, one is not in the interval
of convergence for this series, but hey, trust me.
You can trust me.
This actually does converge.
Now, if you believe that, then what happens when you multiply
everything by 1 half?
Well of course, you just multiply each term by 1 half.
That is definitely true.
Now, if we spread those terms out a little bit and
add everything together term wise. What happens?
Well, we get 1, negative 1 half plus 1 half
is 0. We pull down the 1 third.
Negative 1 4th minus 1 4th, that's minus 1 half.
Pull down the 1 5th. The 1 6ths cancel.
Pull down the 1 7th, etcetera.
And I think you can keep going with this. Some of the terms cancel.
Some of the terms add together.
What is this series. well, it converges, and it converges
as it must to 3 halves of 2. That is, log 2 plus 1 half log 2.
This is true. Trust me.
But at this point, we grow quite concern. Because if we rearrange the terms,
we get exactly what we started with. It is a fact,
that by rearranging the terms in the
alternating harmonic series, we obtain a different answer.
That means your intuition is not so good all the time.
So, what, what can you trust?
And you can't even trust me, sometimes I make mistakes.
There's one thing you can trust and that is LOGIC.
Careful deductive reasoning. We re going
to use tests for convergence or divergence that
are based on logic and calculus. That is what you can trust.
in general, the way that this is going to to work is as follows.
You'll start with some given sequence, and you want to determine whether
the series converges or diverges.
You'll have several and about a half dozen or more tests to choose from.
Not all tests apply to all series.
So you check to see whether a given test applies.
If it doesn't, choose another. If it does, then apply that test.
Some tests don't work with a given series,
in which case you try again.
But if the test does work, then you're done.
You've determined convergence or divergence.
Unfortunately, you might run out of tests, in which case, you fail.
But, that won't happen. We're going to have lots of tests.
Here's the first one. This is called
the nth term test, and it goes as follows. It's an if then
statement. If the hypotheses are satisfied,
then the conclusion follows. Here's the hypothesis.
If the limit as n goes to infinity of a sub n is not equal to 0, then
the series diverges.
That's it. Very, very simple.
Now, in terms of applicability, this is a four star test.
It applies to any series.
No problem. It's easy to use.
You just take the limit of the nth term as n goes to infinity,
and check whether it's zero or not. But in terms of overall usefulness,
well, this is very limited.
In the sense that it only tells you one thing, whether the series diverges.
If that limit is equal to 0, you don't know anything.
What do I mean by that? Let's look at some examples.
If we consider the sum from 0 to infinity of 4 to the
n minus 3 to the n over 4 to the n plus 2 to
the n. Let's check it.
The limit as n goes to infinity of the nth
term is, after a little bit of factoring, the limit
as n goes to infinity of 1 minus 3 quarters
to the n over 1 minus 1 half to the n.
Well, 3 quarters to the n and 1 half to the n
both get very, very close to 0. Therefore this limit is 1.
Which is decidedly non zero.
Therefore this series diverges according to the nth term test.
So far so good.
What about the series then goes from zero
to infinity negative 1 to the n over n factorial.
Well, if we take the limit of a sub n as n goes to infinity, we get
either plus or minus 1 in the numerator and n factorial in the denominator.
That is definitely zero.
And indeed, this series converges.
We know, that this is e to the x, 4x equals negative 1.
Simple enough.
But what about the harmonic series, the sum of 1 over n.
What if we take the limit as n goes to infinity,
we get the limit of 1 over n, that's equal to 0.
But wait a minute, we already know that this series diverges.
What is wrong, has our test failed?
No, our test is not failed. Your logic has failed.
If you have
a logical statement of the form, if p then
q, then you cannot conclude if q then p. That
is called the converse. And the converse does not always hold.
In our nth term test, we have the limit as n goes
to infinity of a sub n non zero, implies divergence of the series.
Divervegence of the series
does not imply that the limit of the nth term is non zero.
The converse does not hold. What then does hold?
What can be said? What can be said is the contrapositive.
P implies q is the same thing as saying that not q implies not p.
We have to negate both statements and
then reverse the arrow. That contrapositive
is always true. So, what we can say is that, if the series
converges, the negation of divergence, then the limit, the nth term is zero.
The contrapositive will hold.
So, if you follow the rules of logic, you won't
get lost. Or, if you do get lost, it's because
you've not followed the rules of logic. In our next two lessons, we'll begin
introducing specific tests for measuring convergence and divergence of
infinite series.
