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

I want to know where a sequence is heading.

[SOUND] Let's think about a specific sequence.

Here's an example to think about.

The sequence a sub n defined by this formula.

The numerators 6n + 2, and the denominators 3n + 3.

We can list off the first few terms.

For instance, if I wanna compute a sub 1, I just plug in 1 for n.

I compute 6 x 1 + 2, divided by 3 x 1 + 3.

That numerator is 8, that denominator is 6, and

I can rewrite eight-sixths as four-thirds, in lowest terms.

And I can keep on going.

By plugging in 2, I find that a sub 2 is fourteen-ninths,

a sub 3 is five-thirds, a sub 4 is twenty six-fifteenths,

a sub 5 is sixteen-ninths, and I could keep on going.

Those fractions really aren't providing me with much insight.

What's the eventual value of the sequence?

Where is the sequence heading?

Might get a better sense by plugging in a thousand, say, what's the thousandth term?

Well that's 6 x 1000 +

2 over 3 x 1000 + 3,

that's 6002 over 3003.

Well that's awfully close to 2.

I can go even farther out in the sequence.

A sub a million is what?

It's 6 x 1 million + 2

divided by 3 x 1 million + 3 and

that works out to be 6 million and

2 divided by 3 million and 3.

And that's insanely close to two.

All right, the limit of a sub n, as n approaches infinity is 2,

to express this idea.

Okay, so what does that really mean?

Well I promise that a sub n is as close as you want to 2,

provided that n is large enough.

For better or for worse, these things are usually written out with fewer words and

more symbols.

Instead of saying, as close as you want to 2, I'll say that a sub n

is within epsilon, some small positive number, of 2.

And how large is large enough?

Well instead of saying large enough,

I'll just say that n is at least as big as some index big N.

So I can say it like this.

So, for every epsilon greater than 0,

there is an index, big N, so that whenever little n is bigger than or

equal to big N, a sub n is within epsilon of 2.

The idea here is that this epsilon

is measuring how close you want a sub n to be to 2.

And I'm telling you that no matter how close you want a sub n to be to 2, if you

go out far enough in the sequence, all the terms after that are actually that close.

Instead of writing within epsilon of 2,

you'll normally see it with absolute value.

So instead of saying, within epsilon of 2, I'll

say that the distance between a sub n and 2 is less than epsilon.

Put it all together.

The same that the limit of a sub n as n approaches infinity equals 2 means, for

every epsilon greater than 0, no matter how close you want to be to the limiting

value of 2, there's some index, n, so that whenever you're farther out in

the sequence than big N, whenever little n is bigger than or equal to big N, and

the absolute value of a sub n- 2, which is measuring the distance between a sub n and

2, that absolute value is less than epsilon.

Of, course, there's nothing really special about the 2 in this.

So in general, to say that the limit of a sub n = L means that for

every epsilon there's a whole number of big N, so

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

the absolute value of a sub n minus that limiting value L, is less than epsilon.

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