Find the Maclaurin series for the functions (2 + x)/(1 - x)

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 44E
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Find the Maclaurin series for the functions (2 + x)/(1 - x)

Expert Solution
Step 1

Given, 

The function f(x)=2+x1-x

 

Step 2

Formula of Maclaurin series is 

f(x) =f(0)+f'(0)1!x+f''(0)2!x2+f'''(0)3!x3+.  .  . 

First find the derivatives of f(x)=2+x(1-x)

f'(x)=(1-x)(1)-(2+x)(-1)(1-x)2=1-x+2+x(1-x)2=3(1-x)2

f''(x)=ddx3(1-x)2=3ddx1-x2-2=-6(1-x2)-3ddx(1-x2)

f''(x)=ddx3(1-x)2=-6(1-x2)-3(-2x)f''(x)=12x(1-x2)3

f'''(x)=ddx12x(1-x2)3f'''(x)=12×(1-x2)3×12-12x(3(1-x2)2)(-2x)(1-x2)6f'''(x)=12×12-12x2+72x2(1-x2)4f'''(x)=144×1+10x2(1-x2)4

 

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