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Problem 67 Hard Difficulty

Use multiplication of division of power series to find the first three nonzero terms in the Maclaurin series for each function.

$ y = e^{-x^2} \cos x $

Answer

$1-\frac{3 x^{2}}{2}+\frac{25 x^{4}}{24}$

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Video Transcript

So for this problem, we want to use the multiplication of division of power Siri's in order to find the first three, none zero terms in the MacLaurin series. So we're given the function y equals e to the negative X squared coastline X. So we already know that co sign X McLaurin Siri's were given it already in table one. If you want to refer to that and it's negative one to the N of X to the to end time over two n factorial. So with that in mind, we also have e to the negative X squared, Equalling the sum of an equal zero to infinity of negative X squared to the end over in factorial. So with that in mind, another way we could write this would be if we want to rewrite it. We could write it as, um, negative one to the end. Since that's often preferred notation Negative one to the end times X to the two n over n factorial. So that being said, we see that four terms of each will probably be enough. So the first four terms of this will be one minus x squared over two factorial put us back into the boards over four factorial minus X to the sixth over six sectorial. And that obviously keeps going on. But we don't need any more. And then this is gonna be one minus X squared over two factorial plus X to the fourth over two factorial. Um, actually, this will be just X squared, Um, minus X to the sixth over three factorial and then plus go on. So what we'll do is we'll multiply these terms. So what we're gonna end up getting as a result is this right here and then we're gonna multiply this Bye. This right here. So when we do that, what we're gonna end up getting as a result, is simplifying it further. We'll get about one minus X squared over to minus two X squared over two. So these will combine to give us a negative three x squared over two so we can do that right there, and then we'll get plus X to the fourth over 24 plus two x to the fourth over to turning this into 24 will make this into 24. So we'll end up getting 25 x to the fourth over 24 so those would be our first three non zero terms within the MacLaurin series for the specific function.

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