Advanced book on Mathematics Olympiad

554 Real Analysis

1

T

∫T

0

|f(x)|^2 dx=a 02 + 2

∑∞

n= 1

(an^2 +b^2 n).

Our particular function has the Fourier series expansion

f(x)=

1

2 π

∑∞

n=−∞

1

n

cos 2πnx,

and in this case Parseval’s identity reads

∫ 1

0

|f(x)|^2 dx=

1

2 π^2

∑∞

n= 1

1

n^2

.

The left-hand side is

∫ 1

0 |f(x)|

(^2) dx=^1
12 , and the formula follows.
495.This problem uses the Fourier series expansion off(x)=|x|,x ∈[−π, π].A
routine computation yields
|x|=
π
2

−

4

π

∑∞

k= 0

cos( 2 k+ 1 )x

( 2 k+ 1 )^2

, forx∈[−π, π].

Settingx=0, we obtain the identity from the statement.

496.We will use only trigonometric considerations, and compute no integrals. A first
remark is that the function is even, so only terms involving cosines will appear. Using
Euler’s formula

eiα=cosα+isinα

we can transform the identity

∑n

k= 1

e^2 ikx=

e^2 i(n+^1 )x− 1

e^2 ix− 1

into the corresponding identities for the real and imaginary parts:

cos 2x+cos 4x+···+cos 2nx=

sinnxcos(n+ 1 )x

sinx

,

sin 2x+sin 4x+···+sin 2nx=

sinnxsin(n+ 1 )x

sinx

.

These two relate to our function as

sin^2 nx

sin^2 x

=

(

sinnxcos(n+ 1 )x

sinx

) 2

+

(

sinnxsin(n+ 1 )x

sinx

) 2

,