   Chapter 11.10, Problem 43E

Chapter
Section
Textbook Problem

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f ( x ) = sin 2 x   [ H int : Use  sin 2 x = 1 2 ( 1 − cos 2 x ) . ]

To determine

To obtain:

The Maclaurin series for the given function.

Explanation

1) Concept:

The Maclaurin series for the function cosx=n=0(-1)n(x)2n(2n)!=1-x22!+x44!-x66!+,  R=

2) Given:

fx=sin2x

3) Calculation:

Consider the given function.

fx=sin2x

By using sin2x=1-cos2x,  fx  becomes

fx=12(1-cos2x)

The Maclaurin series for the function  cosx  is

n=0(-1)n(x)2n(2n)!

The Maclaurin series for the function  1-cosx  is

1-n=0(-1)n(x)2n(2n)!

Replace x by 2x  in the Maclaurin series for the function  1-cosx,  we get the Maclaurin series for the function 1-cos2x.

=1-n=0-1n2x2n2n!

Multiplying  12   by the Maclaurin series for the function 1-cos2x,  we get the Maclaurin series for the function

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