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Welcome to calculus. I'm professor Ghrist.

Â We're about to begin lecture 50 on infinite series.

Â The next few lessons are at the heart of this chapter

Â and deal with infinite series, a hopefully not too unfamiliar topic.

Â We're going to detail specific examples, and consider

Â convergence and divergence of series, in the same

Â manner that we dealt with convergence and divergence of improper integrals.

Â We continue with our program of building a calculus for sequences.

Â Considering, in this lesson, the analog of the improper integral.

Â These are called infinite series. Something that you've seen before.

Â But now, we're going to treat them a bit more carefully.

Â Now, how was it that we handled improper integrals?

Â Recall, the integral from 1 to infinity of f of x dx

Â was defined in terms of a limit. The limit as t goes to infinity

Â of the finite integral of f from 1 to

Â t. So, in defining what we mean by

Â an infinite series, that is, the sum as n goes from 1 to infinity of a sub n.

Â We'll use the same approach.

Â We can think of this infinite series of being something like a

Â discretization of an improper integral. And so, we can

Â define this sum to be the limit as t goes to infinity of

Â the sum n goes from 1 to t of a sub n.

Â That is, the series is really the limit of the sequence of partial sums.

Â Recall that, when it comes to improper

Â integrals, the central and

Â subtle question is that of convergence or divergence.

Â The same occurs with infinite series.

Â Now, you've seen infinite series all

Â throughout this course, from the very beginning.

Â And I have told you repeatedly, don't worry too much about convergence

Â or divergence, we'll worry about that at the end of the course.

Â Well, it is time to push that button. It is time for all of those

Â worries that have been inside your head to come out, and we will deal with them.

Â We will go through the notion

Â of convergence or divergence, slowly and carefully.

Â Giving you lots of practice, so that you get good at it.

Â Let's consider some examples of series. Some are very easy to understand.

Â If we look at 1 plus 1 over e plus 1 over e squared plus 1 over e cubed, et cetera.

Â Well, we recognize this as a geometric series.

Â So, not only do we know that it converges, we know exactly to what value.

Â On the other hand, we recall that the geometric series does not always converge.

Â For example,

Â if we evaluate the geometric series at negative 1, then,

Â one could try to argue that the appropriate value is

Â 0, or, one could argue that the appropriate convergence

Â is to 1, or even to 1 half. None of these

Â holds. This is a divergent series, because the

Â limit of partial sums does not exist. It oscillates.

Â Now, subseries do converge, but are so subtle in how they do so.

Â They're very difficult to evaluate. The sum from 1 to

Â infinity of 1 over n squared does converge as we'll be able to show soon.

Â But it convereges to

Â a value that is very difficult to determine

Â exactly. And that value is pi squared over 6.

Â Some series are not so easy to figure out. Consider 1 plus

Â 1 half plus 1 3rd plus 1 4th plus 1

Â 5th, et cetera. To what, if anything, does this converge.

Â This last example is a special series that is called the harmonic series.

Â Let's investigate this series a little bit.

Â We'll fire up the computer and see what happens

Â when we add, let's say the first 10,000 terms together.

Â We get a result that seems decidedly finite.

Â And it's about 9.79. But what happens

Â if we say add another 5,000 terms to that. Well,

Â our result is only a little bit bigger. We would

Â guess that this is about to converge, since the next

Â term in the series is 1 over 15,001.

Â That is quite small. But let's just type it.

Â Let's add the first 20,000 terms together.

Â Well, we can see that we're slowing down a bit.

Â We've only gone about 0.3 more, not even that.

Â Still I'm not going to be quite comfortable

Â until I see that it's converged to several

Â decimal places.

Â But this does not appear to be happening even with 25,000 terms.

Â So let's sum up the first 100,000 thousand terms.

Â Surely, surely, if this diverges and goes off to infinity, then we would see it.

Â Well, the answer in this case is just a little bit more than we had before,

Â but not so little as to make me conclude that it converges.

Â So what is happening in this case?

Â Well, we're going to have to use something beyond

Â a calculator to determine convergence or divergence in this case.

Â What we're going to use is our calculus intuition.

Â Harmonic series reminds

Â us of the improper integral, as x goes from 1

Â to infinity of 1 over x dx. Indeed

Â we can consider that sum at series as a

Â discretization of this integral.

Â As a left Riemann sum of this integral with a uniform

Â partition of step size one.

Â Now, we know that the integral of 1 over x as x goes from 1 to infinity diverges.

Â And we can conclude from the geometry of this diagram if nothing else, that

Â the harmonic series is strictly bigger. Therefore, we

Â can conclude that harmonic series diverges.

Â What's wonderful about this is that we viewed our continuous or

Â smooth calculus understanding to conclude things about this

Â discreet calculus. Example, and this discussion raise

Â the question, what can you trust when it comes to determining

Â convergence or divergence? As we've seen, a calculator is not

Â the best tool for determining convergence or divergence.

Â What about your intuition?

Â Maybe you should just get a feel for what works.

Â Well, intuition is also flawed. Let's look at an example.

Â I claim that the series 1 minus a half plus

Â a third minus a fourth, plus a fifth, et cetera,

Â something that we'll call the alternating harmonic series, does converge, and it

Â converges to a value of log of 2. Now, why is that the case?

Â Well, recall that log of 1 plus x is exactly this kind

Â of alternating sum of x to the n over n. If we

Â evaluate this at x equals 1, then we obtain this value.

Â Now, technically speaking, one is not in the interval

Â of convergence for this series, but hey, trust me.

Â You can trust me.

Â This actually does converge.

Â Now, if you believe that, then what happens when you multiply

Â everything by 1 half?

Â Well of course, you just multiply each term by 1 half.

Â That is definitely true.

Â Now, if we spread those terms out a little bit and

Â add everything together term wise. What happens?

Â Well, we get 1, negative 1 half plus 1 half

Â is 0. We pull down the 1 third.

Â Negative 1 4th minus 1 4th, that's minus 1 half.

Â Pull down the 1 5th. The 1 6ths cancel.

Â Pull down the 1 7th, etcetera.

Â And I think you can keep going with this. Some of the terms cancel.

Â Some of the terms add together.

Â What is this series. well, it converges, and it converges

Â as it must to 3 halves of 2. That is, log 2 plus 1 half log 2.

Â This is true. Trust me.

Â But at this point, we grow quite concern. Because if we rearrange the terms,

Â we get exactly what we started with. It is a fact,

Â that by rearranging the terms in the

Â alternating harmonic series, we obtain a different answer.

Â That means your intuition is not so good all the time.

Â So, what, what can you trust?

Â And you can't even trust me, sometimes I make mistakes.

Â There's one thing you can trust and that is LOGIC.

Â Careful deductive reasoning. We re going

Â to use tests for convergence or divergence that

Â are based on logic and calculus. That is what you can trust.

Â in general, the way that this is going to to work is as follows.

Â You'll start with some given sequence, and you want to determine whether

Â the series converges or diverges.

Â You'll have several and about a half dozen or more tests to choose from.

Â Not all tests apply to all series.

Â So you check to see whether a given test applies.

Â If it doesn't, choose another. If it does, then apply that test.

Â Some tests don't work with a given series,

Â in which case you try again.

Â But if the test does work, then you're done.

Â You've determined convergence or divergence.

Â Unfortunately, you might run out of tests, in which case, you fail.

Â But, that won't happen. We're going to have lots of tests.

Â