If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Worked example: power series from cos(x)

AP.CALC:
LIM‑8 (EU)
,
LIM‑8.E (LO)
,
LIM‑8.E.1 (EK)
,
LIM‑8.F (LO)
,
LIM‑8.F.2 (EK)

## Video transcript

so let's see if we can find the McLaurys who look like the Maclaurin series representation of f of X where f of X is equal to X to the third times cosine of x squared I encourage you to pause the video and now try to do and remember the Koren series is just the Taylor series centered at zero and let's say our goal here is the first five non-zero terms of the Maclaurin series representation or Maclaurin series approximation of this so I'm assuming you had paused the video you would attempted to do this and there's a good chance that you might have gotten quite frustrated when you do this because in order to find a Taylor series Maclaurin series we need to find the derivatives of this function and as soon as you start to do that it starts to get painful f prime of X is going to be let's see product rule so it's three x squared times cosine of x squared plus X to the third times the derivative of this thing which is going to be two x times negative sine of X negative sine of x squared so just that was pretty painful but so we're going to get more and more painful as we take the second and third and fourth derivatives we might have to take more because some of these terms might end up being zeros because we want the first five nonzero terms so the second derivative this is going to be painful this is going to be painful and then the third and fourth derivatives are going to be even more painful so what do we do here you could just do this and this evaluate immediate zero and then use those for our coefficients but you're probably guessing correctly that there's an easier way to do this and I will give you a hint we know what the pot what we know what the the Maclaurin series for cosine of X is we've done that in a previous video if you if you want to see that again there's another video look up cosine Taylor series at zero on Khan Academy and you'll find that but we already know from that and this is one of the most famous Maclaurin series we know this let's just say G of X let's say G of X is equal to cosine of X well we know how this what this is like though the Maclaurin series approximation of that it's going to be the one minus x squared over two factorial plus X to the fourth over 4 factorial minus X to the sixth over 6 factorial plus and I could keep going on and on on you kind of see where this is going plus X to the eighth over 8 factorial and it just keeps going - plus on and on and on and on but we wanted first a five terms so this is a start I know we one of the first five terms of this thing but bear with me we'll see how this thing right over here is going to be useful so now that I've given you a little bit of reminder on the Maclaurin series representation of cosine of X my hint to you is can we use this to find the Maclaurin series representation of this right up here and remember all this is this is X to the third times and I'm just rereading it for you this is X to the third times G of x squared so that's a sizable hint I encourage you to pause the video again and see if you can work through this so let me rewrite I'm assuming you had a go at it so let me rewrite what I just wrote so I just I just told you that G of X so or f of X f of X if I want to write it as if I wanted to write it as I guess you could say as a is a function or if I want to construct it using G of X I could rewrite it as X to the third tie instead of cosine of x squared that's the same thing as G of x squared so X to the third times G G of x squared G of x squared this G of X is just cosine G of X squares going to be cosine of x squared and I'm going to multiply that times X to the third well can't we just apply this insight to the approximation and you might be asking that question and my answer to you is yes you absolutely can and notice when you substitute your X is 4x squared you'll just get another polynomial and if you multiply that by X to the third you're just going to get another polynomial and that actually will be that Maclaurin series representation of what we started off with we actually will get the Maclaurin series representation of this thing right over here so we could say that f of X is approximately equal to X to the third times X to the third times let me give myself some space right over here so G of x squared so over here this is approximation for G of X and if we kept going on and on forever it would be a representation of G of X so every place where we see an X let's replace it with an x squared so this is going to be one minus so x squared squared is X to the fourth X to the fourth over two factorial which is really just two but I like to put the factorial there because you see the pattern plus if our X is now x squared x squared to the fourth power is X to the 8th X to the 8th power 4 over 4 factorial minus x squared to the sixth power is X to the 12th over 6 factorial and then plus x squared to the 8th is X to the 16th power over 8 factorial and of course we can keep going on and on and on minus plus but we just care about the first five terms nonzero terms and we're saying this is an approximation anyway and so we can say that this is going to be approximately equal to we distribute the X to the third and I'm going to do it in magenta just for fun so distribute the X to the third we get X to the third minus X to the seventh over two factorial plus X to the eleventh over four factorial minus X to the 15th over six factorial plus X to the nineteenth X to the 19th over eight factorial so that's the first five nine zero terms and we are done and when you actually see what we got you realize it would have taken you forever to do this by I guess you could say by brute force because you would have had to find the 19th derivative of all of this madness but when you realize that hey gee if I can just reexpress this function as essentially a X to a power times something that I know the Maclaurin series of especially well if you view it this way if I can write if I can rewrite let me see in a less confusing way if I can rewrite my function so it is equal to sum actually I can even throw a coefficient here if it's equal to some a X to the N power times some function let me just in a new color times dude purple times G of B be X to some other power where I can fairly easily without too much computation maybe I already even know if I know the Maclaurin series representation of G of X if I know what G of X is going to be then I can do exactly what I did just in this video I find the Maclaurin series represented for G every place where I saw an X I replace it with what I have over here the be X to the M power where M is some some exponent that will give me another polynomial another power series and then I multiply it times a X to the N and that's going to once again give me another power series and that will be the power series for my original function very exciting