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 does the same mean?

[MUSIC]

Equality is a subtle topic.

The idea of 4 could just be conveyed by four dots.

It could also be conveyed by uppercase Roman numerals, lowercase Roman numerals,

or a ton of other symbols that have been used in various times and

places throughout the world and throughout history.

All of these mean the same thing.

They all mean four.

They're all, in some sense, equal.

But that's just equality of symbols, of shapes.

There's something much more subtle happening for sequences.

When are two sequences the same?

Two sequences a and b are equal, they're the same, if they start at the same index,

which I'll call big N, and corresponding terms are the same.

So that a sub n equals b sub n whenever n is bigger than or

equal to that starting index.

Let's see how this works out in practice.

And here's one sequence, a sub n.

Here's the sequence that starts with a 0th term and

is defined by the rule that its nth term is 2 to the n.

And here's another sequence, b sub n.

The sequence b sub n, whose 0th term is defined to be 1 and

subsequent terms will be calculated by referring back to previous terms,

so that the nth term is twice the preceding term.

These two sequences, a sub n and b sub n, they're the same.

They're equal.

They're written down really differently.

This sequence b sub n is defined recursively, and

the sequence a sub n is just defined by a single formula in terms of n.

They both start with a term labeled 0, and corresponding terms have the same value.

a sub 0 is 2 to the 0, using this formula, which is 1.

And that's the same as b sub 0.

b sub 1 is using this recursive formula, twice b sub 0.

b sub 0 is 1, so b sub 1 is 2 times 1, which is 2.

And that's the same as a sub 1, which is, using this formula, 2 to the 1st power.

And that pattern continues.

These two sequences both start with a 0 term.

And each term of a sub n is twice the preceding term,

which is exactly the recursive definition that I'm giving for b.

So a sub 1000 equals b sub 1000.

a sub 1 million equals b sub 1 million.

a sub anything equals b sub the corresponding thing.

So the sequence a sub n and the sequence b sub n,

these two sequences are equal as sequences.

Equality isn't about outside appearances.

It's what's inside that matters.

It's the same for sequences.

Two sequences are equal not if they've got the same outside form,

but if their corresponding terms have the same value.

[SOUND]

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