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452 Answers to Odd-Numbered Exercises

Chapter 1 Miscellaneous Exercises

1.f(x)=

∑∞

n= 1

bnsin(nx),

bn=

{ 0 , neven,

4sin(nα)

παn^2 , nodd.

3. Yes. Asα→0, sin(nα)/nα→1.

5.f(x)=

∑∞

n= 1

bnsin(nπx/a),

bn=π^2 h 2 sin(nn 2 πα)

(

1

α+

1

1 −α

)

.

7. a.bn=0,an=0,a 0 =1;

b.

∑∞

n= 1

bnsin(nπx/a),bn=^2 (^1 −cosnπ(nπ));

c. and d. same as a;

e. same as b;

f.a 0 +

∑∞

n= 1

ancos(nπx/a)+bnsin(nπx/a),

a 0 =^1

2

,an=0,bn=^1 −cos(nπ)

nπ

.

9.f(x)=a 0 +

∑∞

n= 1

ancos(nπx/a)+bnsin(nπx/a),

a 0 =

1

2 a,an=−

2 a( 1 −cos(nπ))

n^2 π^2 ,bn=−

2 acos(nπ)

nπ ,

x=−a, −a/ 2 , 0 , a, 2 a,

sum=a, 0 , 0 , a, 0.

11.f(x)=a 0 +

∑∞

n= 1

ancos(nx),

a 0 =^34 ,an=sin(nnππ/^2 ),

x= 0 ,π/ 2 ,π, 3 π/ 2 , 2 π,

sum= 1 ,

3

4 ,

1

2 ,

3

4 ,^1.