   Chapter 11.R, Problem 48E

Chapter
Section
Textbook Problem

Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for e x , sin x , tan − 1 x , and ln ( 1 + x ) . f ( x ) = tan − 1 ( x 2 )

To determine

To find:

i) The Maclaurin series

ii) The radius of convergence R of fx

Explanation

1) Concept:

i) For a power series   n=0cnx-an, there is a positive number R such that the series converges if x-a<R and diverges if  x-a>R, this number R is called the radius of convergence and the series is centred at  x=a.

ii) The Maclaurin series expansion for  tan-1x,

tan-1x=n=0-1nx2n+12n+1=x-13x3+15x5-17x7, for all x

iii) The ratio test:

If limnan+1an=L<1, then the series an is absolutely convergent, and

if limnan+1an=L>1, then the series diverges.

The radius of convergence by ratio test is denoted as R=1L.

2) Given:

fx=tan-1x2

3) Calculation:

It is given that,

fx=tan-1x2

Replace x by x2 in the Maclaurin series expansion for  tan-1x.

tan-1x2=n=0-1nx22n+12n+1

=n=0-1nx22n+12n+1

=n=0-1nx4n+22n+1

This is the Maclaurin series expansion for  tan-1x2

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