Question
Asked Jul 26, 2019
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Find the power series representation for g centered at 0 by differentiating or integrating the power series for f. Give the interval of convergence for the resulting series
2
f(x)1-2x
g(x)=
1-2x)2
The power series representation for g is
k 1
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Find the power series representation for g centered at 0 by differentiating or integrating the power series for f. Give the interval of convergence for the resulting series 2 f(x)1-2x g(x)= 1-2x)2 The power series representation for g is k 1

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Expert Answer

Step 1

Consider the geometric series,

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1 2u", where |<1 1-u n n-0

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Step 2

Substitute u = 2x in the above series to write the function f(x) in power series representation as follows.

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1 -- Σ 2x), where 2y<1 1-2x 0 f(x)=(2x)" where 2s|<1 f(s) -Σ2ν", where 2<1 The interval of convergence for the series is, | 2x | < 1. 2x|<1 -1<2x 1 2x 1 <- 2 2 2 1 1 2 2 Thus, the interval of convergence is 2 2

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Step 3

Differentiate the function f(x) with r...

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d 1 f'(x)d1-2x. -(1-2x) dx -1-1 =-1(12x)(2) 2 (1-2x) = g(x) Hence, g (x) f(x)

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