   Chapter 11.10, Problem 48E

Chapter
Section
Textbook Problem

Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? f ( x ) = tan − 1 ( x 3 )

To determine

i)

To find:

The Maclaurin series of  f

Explanation

1) Concept:

The Maclaurin series for the function tan-1(x)=n=0(-1)n(x)2n+1(2n+1)=x-x33+x55-x77+

2) Given:

fx=tan-1x3

3) Calculation:

Consider the given function,

fx=tan-1x3

The Maclaurin series for function tan-1x is,

n=0(-1)n(x)2n+1(2n+1)=x-x33+x55-x77+

Replace x by x3 in the Maclaurin series for the function tan-1x,  we get the Maclaurin series for function tan-1x3

n=0(-1)n(x3)2n+1(2n+1)=x3-(x3)33+(x3)5

To determine

ii)

To find:

The radius of convergence of  f

To determine

iii)

To graph:

The f & first few Taylors polynomial on the screen.

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