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.
