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

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

0.9 repeating.

[NOISE]

Here's something that's often confusing.

0.9 repeating, by which I mean 0.999999 just going

on like that forever. That is equal to 1.

Not just close to 1. Not a little bit less than 1.

This is exactly equal to 1.

There's a few different ways to think about this.

For example you might already believe that 0.3 repeating, right.

0.3333 and so on is equal to a third.

Now if I multiply all of this by 3 what happens?

Well if I multiply this by 3 I get 0.9 repeating, right.

If I multiply 0.33333 by 3,

I get 0.99999. But if I multiply a third by 3.

I get 1.

We could also formulate this in terms of a series.

I can instead think about it this way. 0.9 repeating is the sum

n goes from 1 to infinity, of 9 times 10 to the negative n.

Well, why is that? Well, what's the first term here?

What happens when I plug in 1?

That's 9 times 10 to the negative first power.

When I plug in n equals 2, that's 9 times 10 to the negative 2nd power.

When I plug in n equals 3, that's 9 times 10 to the negative 3rd power, and so on.

Well what are these?

Alright, 9 times 10 to the negative 1st power is 9 10ths.

9 times 10 to the

negative 2nd power is 9 100's. 9 times 10 to the negative 3rd power

is 9 1000's and it keeps on going. Well, what's 9 10ths?

That's 0.9. What's 9 100s?

That's 0.09. What's 9 1000's?

That's 0.009.

And it keeps on going.

And when I add these up, well, what I end up with is just a

0.9 from here. This 9 gives me a 9 here.

This 9 gives me a 9 here.

The next term gives me the next 9 and so on.

I end up with 0.9 repeating. We can evaluate that series.

So this series, is 9 times the sum n equals 1 to infinity of 10 to the minus n.

And I can make that look even more like a geometric series.

Its 9 times the sum and goes from 1 to infinity of 1

10th to the nth power. Now how do I evaluate that?

We've got a formula for summing an infinite series like that,

it's a geometric series whose common ratio is between zero and one.

So that converges and its sum is 9 times the first term, is 1 10th

divided by 1 minus 1 10th, and what is that?

That's 9 times a 1 10th over 9 10ths. Well, what's 1 10th over 9 10ths?

I could multiply the top and the bottom by 10.

And I get this is 9 times a 1 9th and 9 times 1 9th is 1.

And this discussion brings up an important point.

Any time that we're writing down decimals, I mean if I were

just make up some numbers like 0.57896 and imagine it keeps on going.

Any time I'm writing down real numbers

like that, I'm secretly writing down a series.

What do I even mean by dot, dot, dot?

I really mean the series.

I mean that this is the sum n goes from 1 to infinity of d sub n times

10 to the minus n. Where these d sub n's are the digits,

alright, d sub 1 is 5, d sub 2 is 7, d sub 3 is 8, and so on.

Alright.

So any time I'm writing down a decimal expansion, I'm secretly

writing down an infinite series, just by adding up all the decimals.

To make it a little bit clearer, let me just write down the first few terms.

Right,

n is equals 1 and d sub 1 is 5, means that this is 0.5, and to 0.5, I'm adding the

n equals 2 term, which is d sub 2 is 7 times 10 to the minus 2, which is 0.07.

And then I add the n equals 3 term, which is 0.008

and then I'll add the n equals four term, which is 0.0009 and so on.

So all of these decimal representations of real numbers are secretly just a series.

So when thinking about real numbers or at least

their decimal representations, we're led naturally to think about series.

[SOUND]

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