Advanced book on Mathematics Olympiad

164 3 Real Analysis

487.Compute the ratio

1 +π

4

5 !+

π^8

9! +

π^12

13 !+···

1

3 !+

π^4

7 !+

π^8

11 !+

π^12

15 !+···

.

488.Fora>0, prove that
∫∞

−∞

e−x

2

cosaxdx=

√

πe−a

(^2) / 4
.
489.Find a quadratic polynomialP(x)with real coefficients such that
∣∣
∣∣P(x)+^1
x− 4

∣∣

∣∣≤ 0. 01 , for allx∈[− 1 , 1 ].

490.Compute to three decimal places

∫ 1

0

cos

√

xdx.

491.Prove that for|x|<1,

arcsinx=

∑∞

k= 0

1

22 k( 2 k+ 1 )

(

2 k

k

)

x^2 k+^1.

492.(a) Prove that for|x|<2,

∑∞

k= 1

1

( 2 k

k

)x^2 k=

x

(

4 arcsin

(x

2

)

+x

√

4 −x^2

)

( 4 −x^2 )

√

4 −x^2

.

(b) Prove the identity

∑∞

k= 1

1

( 2 k

k

)=

2 π

√

3 + 36

27

.

In a different perspective, we have the Fourier series expansions. The Fourier series
allows us to write an arbitrary oscillation as a superposition of sinusoidal oscillations.
Mathematically, a functionf:R→Rthat is continuous and periodic of periodTadmits
a Fourier series expansion

f(x)=a 0 +

∑∞

n= 1

ancos

2 nπ

T

x+

∑∞

n= 1

bnsin

2 nπ

T

x.