Uploaded by TheIntegralCALC on 17.11.2011

Transcript:

Hi, everyone. Welcome back to integralcalc.com. Today we’re going to talk about how to find

the Maclaurin series of the function e to the negative 3x. We’ve got a problem over

here and we’ve got a lot going on on-screen already but e to the -3x is our problem.

And what we’re going to do to find the Maclaurin series of this function is first realize what

the Maclaurin series is. I’ve got a couple formulas going on here and I want to talk

about a few of them. We have the Taylor series formula down here on the lower right and the

reason that I wrote out the Taylor series on a Maclaurin series video is because the

Maclaurin series is a specific or unique instance of the Taylor series. The Maclaurin series

we’ve got the formula for up here. And the reason it’s special is because it’s just

like the Taylor series except that the Maclaurin series is specifically exclusive to whenever

a equals zero. So notice down here in the Taylor series, we’ve got for example the

quantity x minus a to the n. When a equals zero as it does in the Maclaurin series, of

course this minus a is going to go away and we’ll be left with x to the n which is what

we have up here in the Maclaurin series. And then similarly, we’re going to have this

a here that becomes zero up in the Maclaurin series formula. So we just wanted to draw

those couple parallels between a Taylor series formula and a Maclaurin series formula. So

just remember that the Maclaurin series formula is the formula for Taylor series when a is

equal to zero. So what we end up here with the Maclaurin

series formula is f to the n of zero over n factorial (n!) all times x sub n. There

are three famous Maclaurin series formulas that describe specific series. One is e to

the x, one is cos x, and the other is sin x. in this case, we’re going to be dealing

with the specific Maclaurin series formula for the function e to the x because our problem

is e to the -3x. What these three formulas allow you to do is substitute any number for

x. So in this Maclaurin series formula for e to the x, we’ve got this x right here.

What we’re allowed to do is put in anything for x no matter what we have in our exponent.

In this case, we have -3x for our exponent. We can drop that in here and we can substitute

or plug in negative 3x anywhere where we see x in our series. So there’s a lot of flexibility

here that this particular Maclaurin series formula gives you.

So in order to write the Maclaurin series for e to the -3x, all we need to do is follow

this formula right here. So we’re going to have the sum from n equals zero to infinity

of x sub n over n factorial. Well, again we can plug in -3x for x so we’ll get negative

3x to the n all over n factorial. So we can plug that in there. What it also allows us

to do is write out the series here because we’ve given it on the right hand side so

all we have to do is follow this to expand it. So we’ll end up with 1 and then remember,

we’re substituting -3x for x, so when we plug that in here, we’re going to get minus

3x. Then we’ll get plus negative 3x squared divided by 2 factorial plus -3x cubed divided

by 3 factorial, etc. So this is really all we have to do to write

Maclaurin series for the function e to the -3x. If you wanted to, you could simplify

this some more. We could expand negative 3x squared, we would get 9x squared, -3x cubed

would give us -27x cubed. But it’s probably almost cleaner to leave it this way because

we’ve got our consistent negative 3x factor across the series so we can go ahead and leave

it just like this. This is our Maclaurin series expansion here of e to the -3x.

So that’s it. That’s how you right the Maclaurin series for something like e to the

-3x. And again, if your function falls along the perimeters of e to the x, no matter what

that exponent is or cos x or sin x, Maclaurin series gives you a very specific formula to

write the series for that particular function. So that’s it. I hope this video helped you

guys and I will see you in the next one. Bye!

the Maclaurin series of the function e to the negative 3x. We’ve got a problem over

here and we’ve got a lot going on on-screen already but e to the -3x is our problem.

And what we’re going to do to find the Maclaurin series of this function is first realize what

the Maclaurin series is. I’ve got a couple formulas going on here and I want to talk

about a few of them. We have the Taylor series formula down here on the lower right and the

reason that I wrote out the Taylor series on a Maclaurin series video is because the

Maclaurin series is a specific or unique instance of the Taylor series. The Maclaurin series

we’ve got the formula for up here. And the reason it’s special is because it’s just

like the Taylor series except that the Maclaurin series is specifically exclusive to whenever

a equals zero. So notice down here in the Taylor series, we’ve got for example the

quantity x minus a to the n. When a equals zero as it does in the Maclaurin series, of

course this minus a is going to go away and we’ll be left with x to the n which is what

we have up here in the Maclaurin series. And then similarly, we’re going to have this

a here that becomes zero up in the Maclaurin series formula. So we just wanted to draw

those couple parallels between a Taylor series formula and a Maclaurin series formula. So

just remember that the Maclaurin series formula is the formula for Taylor series when a is

equal to zero. So what we end up here with the Maclaurin

series formula is f to the n of zero over n factorial (n!) all times x sub n. There

are three famous Maclaurin series formulas that describe specific series. One is e to

the x, one is cos x, and the other is sin x. in this case, we’re going to be dealing

with the specific Maclaurin series formula for the function e to the x because our problem

is e to the -3x. What these three formulas allow you to do is substitute any number for

x. So in this Maclaurin series formula for e to the x, we’ve got this x right here.

What we’re allowed to do is put in anything for x no matter what we have in our exponent.

In this case, we have -3x for our exponent. We can drop that in here and we can substitute

or plug in negative 3x anywhere where we see x in our series. So there’s a lot of flexibility

here that this particular Maclaurin series formula gives you.

So in order to write the Maclaurin series for e to the -3x, all we need to do is follow

this formula right here. So we’re going to have the sum from n equals zero to infinity

of x sub n over n factorial. Well, again we can plug in -3x for x so we’ll get negative

3x to the n all over n factorial. So we can plug that in there. What it also allows us

to do is write out the series here because we’ve given it on the right hand side so

all we have to do is follow this to expand it. So we’ll end up with 1 and then remember,

we’re substituting -3x for x, so when we plug that in here, we’re going to get minus

3x. Then we’ll get plus negative 3x squared divided by 2 factorial plus -3x cubed divided

by 3 factorial, etc. So this is really all we have to do to write

Maclaurin series for the function e to the -3x. If you wanted to, you could simplify

this some more. We could expand negative 3x squared, we would get 9x squared, -3x cubed

would give us -27x cubed. But it’s probably almost cleaner to leave it this way because

we’ve got our consistent negative 3x factor across the series so we can go ahead and leave

it just like this. This is our Maclaurin series expansion here of e to the -3x.

So that’s it. That’s how you right the Maclaurin series for something like e to the

-3x. And again, if your function falls along the perimeters of e to the x, no matter what

that exponent is or cos x or sin x, Maclaurin series gives you a very specific formula to

write the series for that particular function. So that’s it. I hope this video helped you

guys and I will see you in the next one. Bye!