m252-Ch5_Sect5-5

Hello. Welcome to math to 52 calculus to hear Kenyatta College.

My name is rael oppose and I'm the instructor for the course

In this video. We will take a quick dive into section 5.5

in this section,

we're going to use alternating series test to test an

alternating series to see if it converges or diverges.

And then we're going to estimate the some of the alternative series and

explain the meaning of the absolute convergence

or conditional convergence of a series.

So first

an alternating series, what is an alternating series?

A series alternates if you have

a negative one to some end power.

So either one to the end or one to the N plus one or n minus one.

And it basically means that you're going to have

a sequence of numbers that you're adding and subtracting,

adding, subtracting, adding subtracting.

So you're basically going to be alternating between positive and negative.

And the definition of the alternating series says that

it's really negative one times some positive sequence be.

So something like the Kassian uh,

may alternate signs but may not be alternating signs in a consistent manner.

So we don't really consider that as an alternating series

because just because it's switching from positive negative,

there's nothing consistent about the positive or negative switch.

So convergence of an alternating series tells us that if it converges, well,

there's two things that we have to check,

we have to check that the limit goes up to infinity of the sequence of B.

The sequence of numbers B Sub N.

And also that it's going to be uh strictly decreasing in approaching zero.

So here we want to check if it converges,

we can take a look at two consecutive numbers and compare them which one is greater.

So if the larger end is going to be less than the smaller and and uh we take

the limit as n goes to zero by just ignoring the negative one to the n power.

Uh then that is going to make that series converge.

So in our second example, we have an alternating series of n divided by N plus one.

Now, if you take that limit as N divided by N plus one, as N goes to infinity

and that's going to go to one which is not zero.

And so because of that it's going to diverge.

That's kind of like the divergence test.

Uh And so the the alternating series itself will diverge.

Now the remainder of the alternating series test tells us that um we

can see it says here the theorem considering the alternating series test,

um that satisfies the hypotheses for the alternating series test.

So assuming that it actually converges

and we'll let s denote the sum of the series and sn is the f partial sum,

then the remainder is going to be the difference between S and N.

Or S and S sub and the and the partial sum

and that difference is going to be dependent on

the end term the capital and term of the B

sequence that we're looking at or the capital and plus

first term of the B sequence that we're looking at

and so um

now the last objective here is looking

at absolute convergence versus uh conditional convergence.

Now it turns out that if you take the absolute

value of whatever series you're looking at the alternating series,

uh and it converges, then we can say that that series is absolutely convergent.

Now it turns out that if the absolute value does not converge,

but the alternating series does converge,

then we can say that um the series is conditionally convergent.

So those are a couple of definitions that are probably

important to be able to distinguish between the two.

So here are the definitions of absolute convergence and conditional convergence.

And again,

this will be important for us when we start

studying power series because knowing absolute convergence uh is going

to be important for us so that it'll kind of trim our work down just a little bit.

So here are a couple of examples if we have um

an alternating series of something over three N plus one.

In this first series example we have,

if we take the absolute value of that and get rid of the negative one,

then we're looking at the series 1/3 N plus one.

Now we know by some sort of comparison tests

and we can compare that to the harmonic series

that it's gonna diverge.

So the series itself, the series, the absolute value of the series diverges.

Uh So that does not absolutely converge.

But now since the series is actually an alternating series,

this might still have a chance of converging.

And so if we put the alternating series

through the alternating series test which is to

compare two consecutive terms and take the limit as uh and goes off to infinity.

We can see in this line that it actually does converge

but

the alternating series converges and the absolute value series does not converge.

So that's the case where we can say that the series is conditionally convergent

or converges conditionally

in our second example co sign of N divided by N squared if you take the

absolute value that we know that the coastline is going to be a number between um

uh negative one and one. And we know that it changes sign.

So it's not really an alternating series but

we know that it's sometimes it's positive,

sometimes it's negative but if you take the absolute value of that

and then compare that with a series like one over

N squared which we know converges because it's a P.

Series and P. Is bigger than bigger than one.

So that is also going to converge.

So Kassian end over end absolute value of that converges.

So that means that the series is absolutely convergent

and it's absolutely convergent means that the series is going to converge.

Alright,

that is our quick overview of the alternating series test

and alternating series and looking at conditional and absolute convergence.

All right, good luck.

m252-Ch5_Sect5-5

Audio TranscriptAuto-generated