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WZ
Numerade Educator

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Problem 22 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) = 1/x,$ $ a = -3 $

Answer

$f(x)=-\sum_{n=0}^{\infty} \frac{(x+3)^{n}}{3^{n+1}} \quad R=3$

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Video Transcript

The problem is finding the Taylor series for Alfa Back Center that has given value of A and finding associated readers of convergence. So first I have negative three is equal to negative 1/3. A promise is a cultural negative one over expire? Uh huh. From prom is equal to two over x cube and F third derivative. This is a cultural negative three factorial over access to the part of four so we can find that the Earth derivative is equal to negative one to the power of and and times minus one factorial over access to the power of em. Here's a plus one, then f n negative three. It's equally true. Next, the one to the power of end times and minus one factorial over negative three to the power of M s one which is equal to negative and minus one factorial over three to the power of plus one. So the Taylor series for half a wax is some from zero to infinity half and derivative at the point. Negative three over and factorial times X plus three to the power of end, which is equal to some from 0 to 20 Negative one over 10 times three to the power of plus one harms X plus three to the power of the pen. Have a compute limit and go straight. Infinity. Absolute value of make 21 over I'm plus one times ready to power of M plus two over negative one over N times three and plus one, which is equal to the limit, and Austria Final T and over three times and plus one, which is equal to 1/3. So the Raiders of Convergence is equal to 1/1 over three, which is equal to three.