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

Why am I dong any of this?

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Sequences are useful for a variety of reasons.

For starters, sequences help us understand repetitive processes, and some of

those repetitive processes are useful if we're trying to compute something.

Well here's an example of such a process.

I'll define a sequence recursively, X sub n plus 1, we want

over X of n, plus X of n over 2.

Maybe I'll start with the first term of this sequence, as just being 1.

Lets start with X of 1 equals 1, and see what we get.

Well X of 2 is 1 over X of 1, which is 1, plus 1 over 2,

that's 3 halves. We could also try to calculate X of 3.

I get that by taking 1 over X of 2.

So 1 over 3 halves, and adding that to 3 halves over 2.

Now to do that calculation, or I can write 1 over this fraction as 2 3rds.

And instead of writing 3 over 2 divided by 2, I'll write that as 3 4ths.

I'll put this over a common denominator of 12.

So 2 3rds

is 8 12ths, and 3 4ths is 9 12ths.

So all together X sub 3 is 17 12ths.

We can compute more terms with the help of a computer.

Here's the X sub 2 term that we just calculated.

You can compute the next term, the X and 3 term is 17 12ths.

The X and 4 terms is 577, 408ths.

Here's the X and

5 term, in which you'll notice is that these terms

are getting closer and closer to the square root of 2.

Even X of 3, which is 17 12ths, is close to the square root of 2.

Lets see how that works.

So I want to try to convince you that

17 12ths, is approximately the square root of 2.

Well if I square both sides what do I get?

I'm getting that 17 squared divided by 12 squared.

Should be approximately 2. And what's 17 squared?

Well 17 squared is 289 and 12 squared is 144.

And is 289 over 144 close to 2?

Yeah, because the numerator is just about twice the denominator.

So one way to think about the square root of 2, is by thinking about the sequence.

If you want to find a really good approximation of the square root of

2, all you've gotta do is go far enough out in this sequence.

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