Maclaurin Series

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!