m252-Overview

Hello and welcome to math to 52 calculus. To hear kenyatta College.

My name is Ray La Paz and I'm the instructor for the course

In this video. We will take a quick dive into chapter five

sequences and series.

So this chapter will have six sections.

We're gonna first talk about sequences and then series and then go over a

bunch of tests to see whether a sequence or a series converges or diverges

in the first section will get introduced to the idea of sequence is a sequence is

essentially a set of numbers that have some sort of a pattern that you would follow.

Now the curious question is to figure out whether this set of

numbers will eventually settle down to a particular real number value or

not.

And so we're going to explore the idea of converging and diverging sequences

And here is a visual of just plotting a bunch of numbers to

see what happens as you go further and further to the right.

Well, usually you were usually also going to mark this with an index end

to tell you which

term in the sequence you're on. So we have a capital N.

That would be a place where you can start measuring from there and say from capital N.

It's going to be bounded between these two points.

And so you can make that that boundary smaller and smaller.

And then maybe you might have to adjust your end larger and larger but

if you can capture the rest of the points to the right of that,

then you have a convergence sequence

in the next section, we're going to study series.

So instead of looking at numbers individually and plotting them,

we're gonna look at numbers that we're going to add up.

So you have a sequence of numbers that you're going to add up

and then we're going to see if that's some

is going to converge or diverge

and so we're going to get introduced to the idea of the n th partial sum.

And for example, this visual here would be kind of like Sir Pinsky's triangle,

where we would find the sum of the areas of the shaded regions.

And so if we keep shading, if we keep breaking this down to smaller and smaller pieces

and we add up all the sum of all those areas,

are we going to eventually converge or diverge?

Now you would think if you are adding up a whole bunch of numbers

that

it's just going to get larger and larger,

but

that's not necessarily the case.

Take

the series of one over

um

2 to the 10th power.

So

what that is uh is um adding up is the sums of haves and you take halves of what you have,

so imagine

uh being one m away from the wall and say you're gonna walk halfway uh to the wall

and so that's the one half and then now you have one half left over,

you're gonna walk halfway to the wall again,

so you're going to add another 1/4 and then you're gonna walk halfway again.

So that's you're gonna add up another 1/8. And then you keep going.

But if you keep saying the word, I'm only gonna go halfway,

you're not going to go all the way to the wall

and so you're only going to

theoretically

reach the wall

by doing this an infinite number of times.

And so this would be an example where you would add up an

infinite number of numbers

and still get a finite number.

Uh in section uh 5.3 We're gonna start looking at

tests to see whether a particular series converges or diverges

In 5.3 we're gonna take a look at the divergence test.

And uh the divergence test states that if you have

a series and you take just the sequence of numbers

and those sequence of numbers don't approach, zero

then your series itself is going to diverge.

Now there's a little twist to this particular. Um See particular test in that

if if the test does go to zero,

you're still not going to know whether it converges or diverges.

So the only thing that this is definite about is uh

is that if it does not go to zero then the series itself is going to diverge

that's why it's called the divergence test.

And then this visual here takes a look at the integral test.

So the series one divided by N squared would be adding up a bunch of these blocks.

But if you take the integral test,

uh you would compare that to the actual area under the curve

of one over X squared from one going off to infinity.

If you remember

in the integral uh that would be an

improper integral if you integrate from one to infinity

and some of those improper integral converge and diverge.

And so the same thing is going to happen

to their uh sequences that are kind of equivalent

in the next section, We're going to do a comparison test.

So if we're trying to figure out whether a series converges or diverges,

we might want to compare it to a series that we know

convergence and divergence.

And so there's a few different comparisons.

If you have a series that you don't know about and it's

always going to be less than a series that you know converges,

then

you're gonna have to

conclude that the series that you didn't know about also converges.

If you have a series that's larger than another series that you know diverges, then

the series might also diverge.

Now there's this other test called the limit

comparison test where you would take the ratio of

two particular series and if that ratio the limit

of that ratio goes to a particular number,

uh then both series are going to do the same thing.

Either both converge or both diverge.

So we'll get to the details When we explore that particular section,

the next section looks at alternating series test.

An alternating series is a series where you add a term and then you

subtract another term and then you add a term and subtract another term.

So as you could probably guess, this has more of a chance of converging.

And so this has its own set of special rules of convergence and divergence.

And so the alternating series has its own section that we're going to study.

And then in our last section in this chapter

we're going to take a look at the ratio. And the root test

here is the ratio test.

The ratio test is really looking at a ratio of two consecutive terms in your series.

And then you take that limit as and goes off to infinity.

Now if that produces a value that's between zero and one, then it converges.

If it produces a value that's bigger than one, it's it diverges.

And if producing if produces a value that's equal to one,

then we don't know if it converges or diverges.

So we'll have to use a different test outside from the ratio test.

The root test is similar.

Uh And we'll see it when we take a deeper dive into this particular section.

But the ratio test and the root test will give the same results.

So if something doesn't work for the ratio test

uh then it's not gonna work for the root test as well.

So you'll have to refer back to another test that we've learned.

Okay,

so that's it.

This is our introduction to our infinite series and sequences,

and this will get us going for the next chapter.

So,

good luck.

m252-Overview_Ch5

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