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

How large is large enough?

[SOUND]

The definition of limit, says that to get within the

epsilon of L, I just have to go past the big Nth term.

To guarantee that you're within epsilon, how big does N need to be?

We can actually compute this, in some cases.

Consider the sequence

a sub n equals n plus 1 divided by n plus 2.

So what's the limit?

The limit of this sequence as n approaches infinity is 1.

Let me draw a picture of this.

So here' s a number line.

let's put 0 all the way over here, let's put 1 right here.

And I've got a whole bunch of terms on the sequence.

Right? So here's the first term.

Here's the second term. Here's the third term, and so on.

And as I go out further and further in the sequence, the terms get closer

and closer to 1. And the question is, how far do I have to

go out in the sequence, to guarantee that I'm within some epsilon of 1.

Now let's suppose that I want to be within

a 100th of 1. How big does N have to be?

So what I want to do, is find a value for big N.

So that whenever little n is bigger than or equal to big N, I get

that the nth term of my sequence is within 100th of my limit 1.

Right?

This is telling me that the nth term is within epsilon.

Epsilon being 1 100th in this case, of my limiting value 1.

But I can rewrite this.

Instead of writing it this way, I could instead write that

a sub n should be between 99 100ths, and a 101ths.

to be between 99 and a 101ths, is exactly

the same thing as being within a 100th of 1.

Now I've got a formula for a sub n.

So I could instead

write, this is 99 over 100, the formula for a sub n, is n plus

1 over n plus 2, is less than 101ths. So, what I'm trying to

do, is figure out who big I need big N to be, so that whenever little n

is bigger than big N, I know that both of these inequalities hold.

Meaning that my nth term, is really within

a 100th of 1.

Well, one of these inequalities come for free.

This inequality here comes for free, because n plus

1 over n plus 2, is always less than 1.

Right?

The numerator here is smaller than the denominator.

So this thing being less than 1, in

particular, this thing is less than 101 over 100.

So I get this inequality for free.

This inequality, however, requires a little bit of work.

I could solve here by say multiplying both sides by n plus 2 and

by 100, and I end up finding that n needs to be at least 98.

So as long as I choose a value for big N,

which is bigger than 98, that guarantees that this inequality holds.

This inequality holds automatically.

That tells me that my nth term, is really within

a 100th of 1. This is pretty awesome.

Alright, it's pretty cool that we can tell if your past, the 98th term in this

sequence, then you're within a 100th of 1. And there's nothing special about a 100th.

If you wanted to be within a billionth of 1,

you just have to go much further out in the sequence.

And then, you get there.

Right?

And no matter how close you want to be to 1, if you go far enough out in the

sequence, you'll be that close.

And that's exactly what it means, to say that the limit of this sequence is 1.

[SOUND]

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