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Problem 6 Easy Difficulty

Use the definition of a Taylor series to find the first four nonzero terms of the series for $ f(x) $ centered at the given value of $ a. $

$ f(x) = \frac {1}{1 + x}, $ $ a = 2 $ $

Answer

First four non-zero terms of the Taylor series are:
$\frac{1}{3},-\frac{1}{9}(x-2), \frac{1}{27}(x-2)^{2},-\frac{1}{81}(x-2)^{3}$

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Video Transcript

So we want to determine the first four non zero terms of the tailor Siri's, which we ultimately know that but we'll have as a result, is the first term of our Taylor Siri's. We know that a tailor Siri's, is F to then on minus one times to I'm X minus two m s f to the and minus one. Crime of to I am inspects minus two to the end, minus one all over are going to be over and minus one factorial. So, with this in mind, um, what we'll do is plug in different values. So the first term that we have, I will be a zero. So overall one here. So we just have f of to, um f of two X minus 20 So it's just going to give us one so f of to over zero factorial. We know that zero factorial is just one and f of to is one third. So we get one third I'm to or one third, um, over one, which is just going to give us one third. And for the second term, what we'll have is we know that f prime of to since that will have or give us a negative 1/9 IMEs. In this case, we have X minus two. So X minus two to the one. Yeah, and this will be just one factorial down here, so I end up getting as a result is going to be equal. Thio a negative 1/9 times X minus two. And for our third term of the tailor Siri's. What we have is this will be f double crime until which we know to be to over 27. Then times X minus two squared. We know that this will be two factorial because it's three minus one so two factorial and that's going to end up giving us one over seven because the two factorial is just 2 1/27 times X minus two spread. Then lastly, we have our final Taylor. Siri's a term. This will be asked triple prime few, which we know is a negative to over 27. You know, this will be three factorial. This will be X minus two cubed three. Factorial is six. So what we'll end up getting as a result is going to be negative. 1/81 x minus two. Cute. I've just come from simplifying this portion right here. Six divided by that right there. So this will become our final answer that we have for the third term. And these are other subsequent terms that we have.

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