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

We should introduce a bit of terminology.

[SOUND].

What's an arithmetic progression?

An arithmetic progression is a sequence

with a common difference between the terms.

We should see an example.

This might begin 5, 12, 19, 26,

33, and then keep on going. The general

form for the nth term will be 5 plus 7n.

Why is that an arithmetic progression? Well the difference

between each term to the next is seven. Right?

5 plus 7 is 12 12 plus 7 is 19 19 plus 7 is 26 26 plus 7 is 33, and so on.

We can write down a general formula for an arithmetic progression.

Well, here we go.

In general,

a sub n equals some starting number a sub zero

plus a common difference times n. Well, here's another question.

Why are these things even called arithmetic progressions?

Each term Is the arithmetic mean

of its neighbors. For

reals, let's look back at that example. Yeah, in our

example, 12 is the arithmetic mean or the average of 5 and 19.

So 5 plus 19 is 24, and 24 divided by 2 is 12.

Same goes for 19, 19 is the arithmetic mean or the average of its neighbours.

I'm going to write that down, alright, 19 is the average of 12 and

26. What's 12 plus 26?

It's 38. And 38 divided

by 2 is indeed 19.

[SOUND]

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