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!