Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

Loading...

From the course by The Ohio State University

Calculus Two: Sequences and Series

982 ratings

Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

From the lesson

Series

In this second module, we introduce the second main topic of study: series. Intuitively, a "series" is what you get when you add up the terms of a sequence, in the order that they are presented. A key example is a "geometric series" like the sum of one-half, one-fourth, one-eighth, one-sixteenth, and so on.
We'll be focusing on series for the rest of the course, so if you find things confusing, there is a lot of time to catch up. Let me also warn you that the material may feel rather abstract. If you ever feel lost, let me reassure you by pointing out that the next module will present additional concrete examples.

- Jim Fowler, PhDProfessor

Mathematics

Even nonsense can be meaningful.

[SOUND]

Mathematics is more than just things free of inconsistency.

It's more than just that which is the case.

Sometime we're confronted with things, that are really not

nonsensical or things that are just flat out wrong.

And when we're confronted with things like that, we should

really have a feeling or a need to salvage the situation.

We should take the nonsensical or wrong thing and try to salvage it.

Try to think

of some sense in which it might make sense.

So what's the sum n goes from 0 to infinity of 9 times 10 to the nth power?

The party-pooper simply says, the series diverges.

And the party-pooper is right.

This series diverges. And why does it diverge?

Well, it's a geometric series with

R equals 10 and that counter ratio is

bigger than 1. But let's try to salvage the nonsense.

Well, remember back how we looked at the sum N goes from 1 to infinity

of 9 times 10 to the negative nth power and this was, well, it's

equal to 1, but we could also write it as 0.9999 where the

9s keep on going that way.

Well, in this situation, contemplating is really the opposite.

I'm thinking of the sum N goes from 0 to infinity, of 9 times 10 to the N.

What is that?

When I plug it in N equals 0, I get nine time one which is 9.

When I plug it in n equals 1, I get 9 times 10 which is 9.

When I plug in N equals 2, I get 9 times 100 which is 900.

And I'd keep on going. So one way to write this might be, well at

first 9, 9 plus 90, that's 99, 99 plus 900 that's 999.

It's as if I could write it with 9's going this way.

And what if we add 1 to that.

Well, I mean, it's not really a number, right?

But I can still write it down.

So I'll write down

[LAUGH]

a number that's just 9s going this way. And then I'm going to add 1 to that number

[LAUGH].

What do I get? Well, 9 plus 1 is 10, carry the 1.

9 plus 1 is 10, carry the 1. 9 plus 1 is 10, carry the 1.

9 plus 1 is 10, carry the 1. 9 plus 1 is 10, carry the 1.

And I'm going to keep on doing that, right?

So it looks like this thing, which is 9's all the way that way, plus

1, is 0. So whatever this thing is, its a thing

that if i add 1 to it, I get 0. So what is 999 with dots all that way,

its sort of equal and negative 1.

because negative 1 is a thing I can add 1 to to

get 0 and when I added 1 to this thing, I got 0.

Sort of, makes sense.

So if I just ignore convergence altogether, what would it tell me to do?

I'm trying to evaluate the sum and goes from 0 to infinity of 9 times 10 to the N.

Well that would be 9 times the sum N

goes from 0 to infinity of 10 to the N.

This is a, is a divergent geometric series.

Let's just pretend the formula still worked.

How I calculated the value of that series.

That'd be 9 times and the formula for this is

1 over 1 minus the common ratio, which is 10.

Of course, that formula's only valid if it, if it

were convergent series and it's not, but let's just pretend.

What is this? Is this 9 times 1 over 1 minus 10?

That's 9 times 1 over minus 9. Well, 9 times 1 over minus 9 is minus 1.

Which is sort of what we're seeing here, right?

I mean, if we're just ignoring convergence, it looks

like it's telling us that the value of this series.

I mean, it doesn't have a value because it diverges, but

if the value of the series, maybe should be minus 1.

And that, that's wrong because the series diverges.

But, that's maybe the best of the wrong answers.

But what we can keep going with this.

For example, let's do some more calculations.

So let's start with this werid number, which is 9's all that way.

Not really a number, but there we go and let's multiply this by 5.

What do I get?

5 times 9 is 45. So put the 5 down there and carry the 4.

5 times 9 is 45 plus 4 is 49. We have 4 to carry.

5 times 9 is 45 plus 4 is 49.

Got a 9 there and I gotta carry the 4. 5 times nine is 45 plus 4 is 49.

You get the idea.

I'm going to keep on getting 9's that anyway.

So 9, 9, 9, 9 times 5 is 5, 9, 9, 9, 9.

What is this? What happens if I add 5 to it?

5 plus 5 is 0, well it's 10 but I gotta carry the 1.

9 plus 1 is 10, so I put down a 0 and I carry the 1.

9 plus 1 is 10 so I put down the 0 and carry the 1.

All right, so 5 with a bunch of 9's here plus 5 is 0.

So maybe then this number looks to be a lot like a negative 5.

because negative 5 is a thing I can add to 5 to get 0.

And we already saw that 9 is this way looked a lot like negative 1.

So it seems like I've taken a number

that's playing a role sort of like negative 1

and I've multiplied it by 5 and I've got

a number that's playing the role of negative 5.

Which I could tell because when I added 5 to it,

I got back to 0 and that worked better than it should've.

What if we took 999999 and multiplied it by itself?

So I've got

9's all the way to the left, times itself. 9's all the way to the left.

And what is that product? Well, 9 times 9 is 81.

So put the 1 down there and the 8 up there for the carry.

9 times 9 in 81 plus 8, which is 89, so put a 9 there.

I've gotta carry this 8 now.

9 times 9 is 81, plus 8 is 89, so

I'll write the 9 down there and I've gotta carry

the 8.

Now, I keep on carrying the 8's along the top

and I'm going to keep on writing 9's along the bottom.

So it looks like working on the first digit here 9's all the way to

the left times 9 is just one with 9's all the way to the left.

But that's just working on this first digit, now I've

got to move to the next digit on the bottom.

So I'll put a 0 there, and it'll be 9 times 9 is 81, so I put the 8

up there and the 1 down there.

9 times 9 plus 8 is 89, so I'll put the 9 down there and then the 8 up there.

9 times 9 is 81 plus 8, which is 89, so I'll put the 9 down

there and the 8 up there, and I'll keep on going the same, exact way, all right.

So working on that second digit, I've got a 0, a 1 and then all 9's.

And I gotta work on the third digit on the bottom.

Put down two 0's here and then it's exactly the same pattern.

It's going to be 1 followed by 9's going on forever.

And then to work on the next digit on the bottom, it will be

three 0's and then the same patter of 1 followed by a bunch of 9's.

And it's going to keep on going like that.

Well, after I do all the digits along the bottom

in this algorithm I'm supposed to add' them all up.

So now I've got 1.

9 plus 1 is 10, so I put the 0 there and I carry the 1.

1 plus 9 plus 9 plus 1 is 20.

So that's 0 here and I gotta carry a 2 there.

2 plus 9 plus nine plus 9 plus 1, well that's 30.

So I put the 0 there and I'll put a 3 here.

And I'm going to keep on going like that.

And I'm going to get 0's all the way across here.

So what just happened?

Well, it looks like we started with a number that's playing the role of minus 1.

And we multiplied by a number that's playing the role of minus 1.

And when I actually did that multiplication,

the answer that I ended up getting looks to be 1.

Which is just what I hope it would be.

So it really is seeming like it's working.

And when we started out with

this nonsensical thing right, this divergent series.

But we're sort of taking the nonsense seriously, we've ended up

getting a system that works sort of like the negative numbers.

And in fact computers do store negative numbers this way.

Of course, computers usually don't

work base 10, they work base 2 and when you work

base 2, you do the same kind of game called two's complement arithmetic.

And this sort of thing comes up in just pure mathematics.

This sort of thing is often called the P-adic numbers

and in particular when we're using powers of ten, 10-adic numbers.

So mathematics is so powerful, that even nonsense can lead to a reasonable fury.

It's one thing to win battles, just by force alone, right?

To be great, by virtue of simply being correct all of the time.

But it's something else entirely, to win those battles, when you're weak.

To be on the right track, even when you're entirely wrong.

And I think the fact that mathematics works

like this really shows that there's something to it.

It's not just symbols that we're pushing around on a page.

We're really out there,

exploring something that's really there, something really beautiful

and we're just fortunate to get to take part.

[SOUND]

Coursera provides universal access to the world’s best education,
partnering with top universities and organizations to offer courses online.