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

What are sequences?

[MUSIC]

A sequence is a list of numbers. For example here is a sequence.

This is a sequence that goes to one, one, two, three, five, eight

and I'll write dot, dot, dot to remind me the sequence goes on forever.

It's an unending list of numbers. Each of these numbers in the list

is referred to as a term in the sequence. I want some notation so I

can refer to specific numbers in the list.

I'll write a sub one for the first term in my sequence,

a sub two for the next term, a sub three for the next

term, a sub four for the next term, a sub five for

the next term, a sub six for the next term, and so on.

So in this particular example, I could say that the

sixth term in my sequence is eight, or the third

term in my sequence is two.

If I want to talk about the sequence as a whole,

I'll perhaps just write down a sub n, maybe in parentheses.

And I'll use this notation to talk about the entire sequence.

But by plugging in different values for n, I

can then speak of specific terms in the sequence.

This means that a sequence really amounts to some assignment from

these indices, the subscripts, to other real numbers, and a gadget that

assigns numbers to other numbers, it's a function, so yeah the sequence

a sub n is secretly a function, f of n, because both

of these gadgets are just ways of assigning real numbers to other numbers.

In this case, the sequence assigns a real number to these ns, to one, two, three,

and so on.

So it's worth pointing out then that what's really the

domain of this function that's playing the role of the sequence?

Well, the domain of f should just be whole numbers.

I don't want to talk about the 5 thirds term of the sequence, right?

It doesn't really make sense, usually, to talk about a sub 5 thirds, or a sub pi.

But it does make sense to talk about a sub

three, and a sub 100.

So the domain of my function, the thing that I'm allowed

to plug in for n, should really just be whole numbers.

And I usually develop that with this fancy-looking N.

I can take a look at a sequence numerically.

I can just take a look at the terms and see what their approximate values are.

Besides thinking numerically, I could also think about a sequence geometrically.

For example,

here's a number line.

Let's say, here is zero, here is one, here is

two, here is three and it would keep on going.

And I could plot the terms of my sequence on this number line.

You could imagine some sequence where the first term is here between zero and

one, the second term is between one and two, may be the third term.

Is here between one and the second term.

Maybe the

fourth term is over here between a sub one and one.

Maybe the fifth term is over here, just a little bit less than three.

You could plot the terms in your sequence to try

to get some idea about what the sequence look like.

I can think about a sequence.

Algebraically, well algebraically I could define a sequence by

giving you a rule to compute the nth term.

May be a sub n is some algebraic

formula like n square plus 1 and in that case I can just compute the first term is

1 squared plus 1 which is 2, the second term Is 2 squared plus 1 which is 5.

The 3rd term is 3 squared plus 1 which is 10, and so on.

Our goal now is to build some interesting

sequences, and explore the properties that those sequences have.

[NOISE]

[NOISE]

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