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?"

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From the course by The Ohio State University

Calculus Two: Sequences and Series

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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

Sequences

Welcome to the course! My name is Jim Fowler, and I am very glad that you are here.
In this first module, we introduce the first topic of study:
sequences. Briefly, a sequence is an unending list of numbers; since a sequence "goes on forever," it isn't enough to just list a few terms: instead, we usually give a rule or a recursive formula.
There are many interesting questions to ask about sequences. One question is whether our list of numbers is getting close to anything in particular; this is the idea behind the limit of a sequence.

- Jim Fowler, PhDProfessor

Mathematics

Let's put all the integers together.

[MUSIC]

The goal here is to come up

with a sequence that mentions every single integer.

But before we do that, let's first try to come

up with a sequence that mentions every non negative integer.

It's the sequence a sub n equals n where the index n starts at zero.

So the terms of the sequence are zero, one, two, three, and so on.

And of course,

I could negate that sequence to get

a sequence that mentions every negative integer.

So I could look at the sequence b sub

n equals negative n, but let's start n at one.

And if I do that, then the terms of this sequence start minus one, minus two,

minus three, minus four and so on. But I want both, I want a single sequence

that includes among it's terms every

positive integer, every negative integer and zero.

Is it even possible?

Yes, I'll, I'll weave together the two sequences that we've already built.

So what do I mean?

Well let's take a look at these two sequences.

I could put them together, right. I could weave them together.

I could start with 0, then do minus 1,

then 1, then minus 2, then 2, then minus 3.

I get

a new sequence that would end up mentioning every single integer, with

start zero, minus one, one. Minus two, two,

minus three, three, minus four, four, and it would keep on going like that.

I'd like a formula for that sequence.

Well here's a formula for the sequence. C sub n will be defined

via this piecewise definition depending on the parity of n, whether n is odd or even.

If n is odd, then the nth term will be negative n plus 1 over 2.

And if n is even, then the nth term is just n over two.

And I'll start with the index zero.

And this will give me this sequence, right, the zero term, when I plug in

zero, zero is even, zero over two is zero, and that gives me the zero.

When I plug in one, one is odd, so I

get negative one plus one over two, that's negative one.

When I plug in two that's even, so it's two over two, that's this one here.

When I plug in three, three's odd, so it's negative three plus one

over two, that's negative two and it just keeps on going like that.

There's another way to think about this.

To say that I've got the same quantity of dots and squares.

Is really to say that there's some method by which I can pair off the

dots and squares so that every square gets a dot and every dot gets a square.

And once I've paired them off like this it's very

believable that there's the same quantity of dots and squares.

Well something similar is going on with non-negative integers and all integers.

If I just take a look at the non negative integers, I perhaps want

to show others the same quantity of non

negative integers as there are just all integers.

And to do that, I just have to tell you some

method for pairing off non negative integers with all the integers.

And that's exactly what this sequence does, right?

It assigns to zero the number zero.

It assigns to one, the number minus one,

it assigns to the index two, the number one,

to the index three, the sequence assigns the number negative two.

To the index four, it assigns the number two.

To the index five, it assigns the number three, to the index

six, it assigns the number three and it will keep on going.

And eventually, right, we've ended up pairing off

every single non negative integer with every single integer.

And that really means that there's the same quantity of non negative integers

as there are all integers.

That should really give you pause, that, that sounds impossible, alright?

Think about some physical object, some finite object like, coffee beans.

If I've got some coffee beans, but then I take some away, now

I've got fewer coffee beans, but the collection of all integers is different.

If I start with all the integers and just take away the

negative integers I've got the same quantity of things.

Why does that work?

Well one definition of an infinite quantity is a quantity

that needn't get smaller, even when you take something away.

[SOUND]

[SOUND]

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