Advanced book on Mathematics Olympiad

Real Analysis 575

Remark.This is a particular case of integrals of the form

∫∞

0

f(ax)−f(bx)
x dx, known as
Froullani integrals. In general, iffis continuous and has finite limit at infinity, the value
of the integral is(f ( 0 )−limx→∞f(x))lnba.

524.We do the proof in the case 0<x<1, since for− 1 <x<0 the proof is completely
analogous, while forx=0 the property is obvious. The functionf:N×[ 0 ,x]→R,
f (n, t)=tn−^1 satisfies the hypothesis of Fubini’s theorem. So integration commutes
with summation:

∑∞

n= 0

∫x

0

tn−^1 dt=

∫x

0

dt

1 −t

.

This implies

∑∞

n= 1

xn

n

=−ln( 1 −x).

Dividing byx, we obtain

∑∞

n= 1

xn−^1

n

=−

1

x

ln( 1 −x).

The right-hand side extends continuously at 0, since limx→ (^01) tln( 1 −t)=−1. Again we
can apply Fubini’s theorem tof (n, t)=t
n− 1
n onN×[^0 ,x]to obtain
∑∞
n= 1
xn
n^2

=

∑∞

n= 1

∫x

0

tn−^1

n

dt=

∫x

0

∑∞

n= 1

tn−^1

n

dt=−

∫x

0

1

t

ln( 1 −t)dt,

as desired.

525.We can apply Tonelli’s theorem to the functionf(x,n)=x (^2) +^1 n 4. Integrating term
by term, we obtain
∫x
0
F(t)dt=
∫x
0

∑∞

n= 1

f (t, n)dt=

∑∞

n= 1

∫x

0

dt

t^2 +n^4

=

∑∞

n= 1

1

n^2

arctan

x

n^2

.

This series is bounded from above by

∑∞

n= 1

1

n^2 =

π^2
6. Hence the summation commutes
with the limit asxtends to infinity. We have

∫∞

0

F(t)dt=xlim→∞

∫x

0

F(t)dt=xlim→∞

∑∞

n= 1

1

n^2

arctan

x

n^2

=

∑∞

n= 1

1

n^2

·

π

2

.