Answered: 3 f(x) х 3 + x

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter6: Fractions

Section6.1: Simplifying Fractions

Problem 31WE

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Question

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

Expert Solution

Step 1

Consider the function $f (x) = \frac{3}{3 + x}$ .

$f (x) = \frac{3}{3 + x} = \frac{3}{3 (1 + \frac{x}{3})} = \frac{1}{(1 + \frac{x}{3})} = {(1 + \frac{x}{3})}^{- 1}$

Since ${(1 + x)}^{- 1} = 1 - x + x^{2} - x^{3} + . . . + {(- 1)}^{n} x^{n} + . . . v a l i d w h e n |x| < 1$ .

$f (x) = {(1 + \frac{x}{3})}^{- 1} = 1 - \frac{x}{3} + {(\frac{x}{3})}^{2} - {(\frac{x}{3})}^{3} + . . . + {(- 1)}^{n} {(\frac{x}{3})}^{n} + . . . v a l i d f o r |\frac{x}{3}| < 1 = 1 - \frac{x}{3} + {\frac{x}{9}}^{2} - {\frac{x}{27}}^{3} + . . . + {(- 1)}^{n} {(\frac{x}{3})}^{n} + . . . t h e e x p a n s i o n v a l i d f o r |x| < 3$

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