   Chapter 11, Problem 61RE

Chapter
Section
Textbook Problem

# Suppose that f ( x ) = ∑ n = 0 ∞ c n x n for all x. (a) If f is an odd function, show that c 0 = c 2 = c 4 = ⋯ = 0 (b) If f is an even function, show that c 1 = c 3 = c 5 = ⋯ = 0

(a)

To determine

To show : If f(x)=n=0cnxn is an odd function, then c0=c2=c4==0 .

Explanation

Proof:

The function f(x)=n=0cnxn can be written as odd and even terms shown below,

f(x)=n=0c2nx2n+n=0c2n+1x2n+1 (1)

If the function f(x) is odd, then f(x)=f(x) .

Substitute x for x in equation (1),

f(x)=n=0c2n(x)2n+n=0c2n+1(x)2n+1=n=0c2n((x)2)n+n=0c2n+1(1)2n+1(x)2n+1=n=0c2n(x2)n+n=0c2n+1((1)2)n(1)(x)2n+1=n=0c2n(x)2nn=0c2n+1(x)2n+1

f(x)=n=0c2n(x)2nn=0c2n+1(x)2n+1 (2)

Substitute the equations (1) and (2) in f(x)=f(x) ,

(b)

To determine

To show : If f(x)=n=0cnxn is an even function, then c1=c3=c5==0 .

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