WEBVTT
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So we're gonna be looking into the McLaurin. Siri's
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using the definition of MacLaurin series. So what we
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have is f of X being equal to e to
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the negative two x So first thing we're gonna wanna
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dio is find some of the first derivatives and calculate
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their values at X equals zero. So obviously f
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of as X equals each of the negative two X
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and f of zero equals one. Then F prime
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of X is equal to a negative to e to
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the negative two x and f prime of zero would
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then give us a negative, too. F double
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Prime of X is going to be for E to
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the negative two x f double prime of zero is
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gonna be four and then we'll do one more.
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We have f triple Prime of X is going to
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equal a negative eight e to the negative two x
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So have triple prime of zero is going to be
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a negative eight. So we see that we can
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plug everything into the General Taylor form with the McLaurin
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being equal zero. So we see that ffx is
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equal to f of a, but the is equal
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to zero. So one minus f prime of a
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over one factorial. So one minus two X and
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then we're just gonna follow the General Taylor Taylor form
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using a being equal to zero. So what we're
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gonna get is, um, this is gonna be
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then plus for over two factorial X squared minus 8/3
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factorial x cubed plus you know, 16/4 factorial X
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to the fourth and so on. Then with that
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, we have that This is the summation from an
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equal zero to infinity of negative one to the end
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because we wanna alternate negative and positive times two to
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the end to get the 248 16 going and then
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times X to the end to get the degree of
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the X value. And that's obviously going to be
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over in factorial. Then by the alternating Siri's test
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, we see that the terms become smaller as an
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increases. Um, and in fact, Toyo grows
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faster than the numerator. So it's going to converge
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for all real numbers. Converges for all real numbers
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. And we see that the radius of convergence is
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r equals infinity. Then we could also do a
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ratio test to test it, but regardless we'll see
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that it does converge and the race of convergence is
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r equals infinity.