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

Finding limits is hard. I wish I could just prove that they exist.

[MUSIC].

Here's a theorem that guarantees a sequence converges.

If the sequence is bounded and monotone, then the limit exists.

Why is this important?

Well, I often can't tell whether a sequence converges.

But I may be able to show that a sequence is both bounded and monotone in

that I know it has a limit.

But why should I believe that the monotone conversions theorem is true?

Well let's think about this geometrically.

Suppose I've got a number line and I've got terms of

my sequence, x of 1, x of 2, x of 3.

Let's pretend they're increasing.

And let's pretend that the sequence is bounded.

So I know that the sequence

never exceeds this value b.

So I've got a sequence which is increasing and bounded above.

Well what can happen, right?

As I go out further and further in that sequence, I can

never pass b and yet I have to keep moving to the right.

So hopefully it seems plausible that with these conditions, this sequence can't

help but converge to some limiting value l.

So hopefully it seems plausible but we don't

yet have the tools to give a formal proof.

[SOUND]

[SOUND]

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