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?"

Loading...

From the course by The Ohio State University

Calculus Two: Sequences and Series

979 ratings

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

Series

In this second module, we introduce the second main topic of study: series. Intuitively, a "series" is what you get when you add up the terms of a sequence, in the order that they are presented. A key example is a "geometric series" like the sum of one-half, one-fourth, one-eighth, one-sixteenth, and so on.
We'll be focusing on series for the rest of the course, so if you find things confusing, there is a lot of time to catch up. Let me also warn you that the material may feel rather abstract. If you ever feel lost, let me reassure you by pointing out that the next module will present additional concrete examples.

- Jim Fowler, PhDProfessor

Mathematics

Let's think about a more complicated series.

[SOUND].

How can we approach this series? It's

the sum, k goes from 0 to infinity, of sine of k, squared, divided by 2 to the k.

Does this series diverge or converge?

Look suspiciously similar to a series that we do understand.

Yeah, it looks

a bit like this series, right?

If I just got rid of this sin of k squared

term and replaced it with a 1, I'd have this geometric series.

This series converges and its value is 2.

Let's be careful, I want a precise relationship between the mysterious series

that I don't understand, and the geometric series that I do understand.

Precisely how are these related?

Well, sine of k is between minus 1 and 1, so sine of k squared is between 0 and 1.

And now if I were to divide all of this by 2 to the k.

Look, I now I have, sine of k squared divided by 2

to the k is between 0 and 1 over 2 to the k.

What's the original goal, what am I actually trying to understand here.

I'm trying to understand whether this series converges or diverges.

What does that question mean?

That question really means, that I'm supposed to be looking at a sequence

of partial sums, should add up the terms between k equals 0 and n.

And then ask about the limit of the partial sum.

And if this limit exists,

well that's exactly what it means to say that this original series converges.

What sort of sequence is S sub n?

One thing I know about sequence of partial

sums is the sequence of partials sums is non-decreasing.

How do I know that?

Well, the terms that I'm adding up look

like this and those terms are non negative.

So, if you add a non-negative number, the result's not getting any lower, all

right?

And that means that the sequence of partial sums is not getting any smaller.

It's non-decreasing.

Not only that, the sequence of partial sums, s of n, is also bounded.

So, why bounded?

Well, it again boils down to this inequality.

Because this is less than this, if I add up a

bunch of these, that's less than adding up a bunch of these.

It's exactly what I'm saying here. If I sum all of these terms,

it's less than or equal to the sum of these terms, because each

of these terms is less than or equal to each of these terms.

And that's just because sine square of k is less than or equal to 1.

But I know something else here, right?

This is a convergent geometric series.

As I let n drift off to infinity, this is approaching 2.

So putting this all together, I've got that these

partial sums are bounded by 2. So I've got a bounded monotone sequence.

But, a bounded monotone sequence converges, so

we know something about the original series.

And because the sequence of partial sums converges, that's

exactly what it means to say that the series converges.

This idea, this idea of taking a series we do understand, and using

that series to explore a series we don't understand.

This idea has a name.

It's called the comparison test. This idea is important enough that, in a

future video, we're going to write down a specific statement of the comparison test.

[SOUND].

Coursera provides universal access to the world’s best education,
partnering with top universities and organizations to offer courses online.