1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Miscellaneous Exercises 133

a. |x|=^1
2

−^4

π^2

∑∞

k= 0

1

( 2 k+ 1 )^2

cos

(

( 2 k+ 1 )πx

)

, − 1 <x<1,

π^2

8

= 1 +^1

9

+^1

25

+···;

b.

4

π

∑∞

k= 0

1

2 k+ 1 sin

(

( 2 k+ 1 )πx

)

=

{ 1 , 0 <x<1,

− 1 , − 1 <x<0,

π

4 =^1 −

1

3 +

1

5 −

1

7 +···;

c. |sin(x)|=π^2 −π^4

∑∞

n= 1

1

4 n^2 − 1 cos(^2 nx),

1

2

=^1

3

+^1

15

+^1

35

+···.