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

Let's think geometrically.

[MUSIC]

We're often exploring the limit of a

sequence by looking at some numeric evidence.

You know, we might evaluate a few hundred terms of

the sequence and notice, we're getting really close to something numerically.

And we might guess that that's the limit.

But we can also think about limits in a, in a more geometric way.

So, here I've got a number line, and I've

plotted the terms in my sequence on that number line.

And, it looks like the limit is L.

But what does that mean?

Well, if I zoom in on the number line a bit, it looks

like all of the terms of my sequence are within a hundredth of L.

As long as I'm after the 53rd term in my sequence.

And, if I

zoom in again. All the terms of my sequence are within a

thousandth of L, as long as I'm after the 181st term in my sequence.

And as close as I want to get to L, I can

do that, as long as I go far enough out in the sequence.

So a limit's really a promise.

And it's a promise that, if you want the terms of your sequence to be close to L,

well you can do that.

You can get the terms to be as close as you want to L, as long as you

throw away enough of the initial terms and just

restrict your attention to the tail of the sequence.

[SOUND]

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