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

There's more terminology.

[SOUND]

[MUSIC]

What is a geometric progression? A geometric progression, is a sequence

with a common ratio between the terms. We should see an example.

Maybe the sequence starts 3, then 6, 12, 24,

48, 96 and it keeps on going. And looks like the

general rule for this sequence is a sub n equals 3 times 2 to the n.

Why is that a geometric progression?

Well, there's a common ration of 2 between each of these terms.

To get from 3 to 6, I have to multiply by 2.

To get from six to 12, I multiply by 2. To get from 12 to 24, I multiply by 2.

To get from 24 to 48, I multiply by 2. Alright?

That's the common ratio between all the terms in this sequence, it's 2.

We can write down, a general formula for a geometric progression.

So I can write a sub n, equals the first term,

A sub 0, times the common ratio R to the nth power.

In this particular example, A sub 0, the first term is 3.

And the common ratio is 2.

Here's a question, why are these things even called geometric progressions?

Well in a geometric progression, each term is the geometric mean of it's neighbors.

Okay, but what is a geometric mean?

Well, the geometric mean of two numbers, of a

and b, is defined to be the square root.

>> Of A times B.

>> Why is a geometric mean, called geometric

at all? What's geometric about it?

>> Well, here's on geometric story you could tell yourself.

You could build a rectangle, one of who's sides

is A, and the other side has length B.

Then this rectangle has area AB.

I'm going to build a square.

And I want to build a square, whose area is also ab.

What's its side length?

Well the side length will be the square root of ab.

So this is some kind of geometric sense, in which an average

of a and b might deserve to be the square root of ab.

A geometric average.

So the deal with geometric progressions, is that

each term is the geometric mean of its neighbors.

So let's see that in our original example: 3, 6, 12, 24, and so on.

The claim is that in a geometric

progression, each term is the geometric mean of

it's neighbors. Let's see that here.

What's the geometric mean of 3 and 12.

Well, it's the square root of 3 times 12, that's the square root

of 36, that's 6 so, yeah, 6 is the geometric mean of it's neighbors.

Let's try to get em 12.

What's the geometric mean of 6 and 24?

Well, that's the square root of 6 times 24.

6 times 24 is 144. And the square root of

144 is 12.

So, yeah, 12 is a geometric mean of 6 and 24.

The limit of a geometric progression,

depends very strongly on that common ratio.

Well in our example here, what's the limit as n approaches infinity of a sub n?

It's infinity, I can make a sub n as big as I like, provided I choose n big enough.

What if the common

ratio were a third? Here's an example of a geometric

progression, with common ratio a third. 1, a 3rd,

a 9th, a 27th, an 81st, and so on.

What's the limit.

Of a sub n in this case as n approaches infinity.

Well that's really

the limit as n approaches infinity of 1/3 to the nth power because that's a formula

for the nth term in this sequence. Well, that limit is 0, right?

By making n big enough, I can make a sub n as close to 0 as I like.

Other interesting things can happen, too.

You should think about what happens, when that common ratio is negative.

[NOISE]

[SOUND]

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