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Problem 26 Medium Difficulty

Find the Taylor series for $ f(x) $ centered at the given value of $ a. $ [Assume that $ f $ has a power series expansion. Do not show that $ R_n (x) \to 0.$] Also find the associated radius of convergence.

$ f(x) = \sqrt x $ $ a = 16 $

Answer

$4+\frac{x-16}{8}+\sum_{n=2}^{\infty}(-1)^{n+1} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-3)}{2^{5 n-2} n !}(x-16)^{n}, \quad R=16$

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Ek

Ella K.

June 2, 2021

solve f(x)= x^4+2 a=-1 Taylor series

Video Transcript

So for this problem 26 we want to find the Taylor series centered at the value of a So in this case, a is 16 and f of X equals the square root of X. So we know right off the bat that f of a is going to equal For now, we want to find some derivatives. So we take the first derivative and what we get is that this is equal to one over two x to the one half f double Prime of X is equal to negative one over two times two x to the three house F triple prime of X is equal to 3/2 times two times two x to the five house. We start to see the pattern emerge and we get f to the fourth. Prime of X is equal to a negative three times five over. Two times, two times, two times two x to the second house. So what we see in terms of F prime of 16, that's going to give us 1/2 times four f double prime of 16 is going to give us negative 1/2 times two times for cubed F triple prime of 16 will give us 3/2 times, two times, two times for the fifth. And then lastly, um, after the fourth prime of 16 is going to give us three times five over two times two times two times, two times for to the seventh. So based on that, we can plug these values into our Taylor series. Um, and since we know how the Taylor series should be structured, what it's going to look like is that f of X is equal to four. Our first term, plus one over, um plus 1/2 times four X minus 16 minus one over two times, two times for cube times, two factorial times x minus 16 squared and that's going to keep going on. We'll show the third term that's going to be plus three over, um two times two times two times four to the fifth times three factorial. And that's gonna be X minus 16. Cute. So it's going to keep going on, so the way that we can, right this is going to be, um, one over four to the two and minus one times and factorial Times X minus 16 to the end. So if we look at the rest of our numbers, what we end up getting when we do, the summation is going to equal four plus our X minus 16/8, plus the sum from an equals two to infinity of negative one to the end, plus one. Because it can be an alternating series, um, of one times, three times five times. And that's going to keep going all the way up until two and minus three, all over two to the five and minus two and factorial and then we have to multiply that right there. Times X minus 16 to the end. We can use a ratio test to find the radius of convergence. So when we take the limit as an approaches infinity, what we end up getting is the limit. As an approaches infinity of two x minus 32 Nice X over N plus 16 over n over 32 times one plus one over n. And that's gonna end up giving us X over 16 minus one. Because as this goes to infinity, this whole thing goes to zero. Same thing here and here. So this is going to be that right there. Which means that our radius of convergence will be zero less than X less than 32. So our radius of convergence is 16.