   Chapter 11.10, Problem 44E

Chapter
Section
Textbook Problem

# Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.44. f ( x ) = { x − sin x x 3 if     x ≠ 0 1 6 if     x = 0

To determine

To obtain: The Maclaurin series for given function f(x) .

Explanation

Given:

The function is f(x)={xsinxx3  if x016             if x=0  .

Result used:

The expansion of sinx is n=0(1)n1(2n+1)!x2n+1 . (1)

The radius of convergence of sinx is .

Calculation:

Consider the function f(x)=xsinxx3 if x0 .

Obtain the series of f(x)=xsinxx3 .

xsinxx3=xn=0(1)n1(2n+1)!x2n+1x3   [sinx=n=0(1)n1(2n+1)!x2n+1]=xx(2(0)+1)!n=1(1)n1(2n+1)!x2n+1x3=xxn=1(1)n1(2n+1)!x2n+1x3=n=1(1)n1(2n+1)!x2n+1x3

Substitute n+1 for n,

xsinxx3=n=0(1)n1(2(n+1)+1)!x2(n+1)+1x3=n=0(1)n1(2n+3)!x3x2n+3=n=0

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