Advanced book on Mathematics Olympiad

178 3 Real Analysis

∫b

a

∑∞

n= 0

f (n, x)=

∑∞

n= 0

∫b

a

f (n, x).

Here we are allowed to commute the sum and the integral if eitherfis a positive function,
or if

∫b

a

∑∞

n= 0 |f (n, x)|(or equivalently

∑∞

n= 0

∫b
a|f (n, x)|) is finite. It is now time for
an application.

Example.Compute the integral

I=

∫∞

0

1

√

x

e−xdx.

Solution.We will replace√^1 xby a Gaussian integral. Note that forx>0,

∫∞

−∞

e−xt

2

dt=

∫∞

−∞

e−(

√xt) 2

dt=

1

√

x

∫∞

−∞

e−u

2

du=

√

π

x

.

Returning to the problem, we are integrating the positive function√^1 xe−x, which is inte-
grable over the positive semiaxis because in a neighborhood of zero it is bounded from
above by√^1 xand in a neighborhood of infinity it is bounded from above bye−x/^2.

Let us consider the two-variable functionf (x, y)=e−xt
2
e−x, which is positive and
integrable overR×( 0 ,∞). Using the above considerations and Tonelli’s theorem, we
can write

I=

∫∞

0

1

√

x

e−xdx=

1

√

π

∫∞

0

∫∞

−∞

e−xt

2

e−xdtdx=

1

√

π

∫∞

−∞

∫∞

0

e−(t

(^2) + 1 )x
dxdt

=

1

√

π

∫∞

−∞

1

t^2 + 1

dt=

π

√

π

=

√

π.

Hence the value of the integral in question isI=

√

π. 

More applications are given below.

521.Leta 1 ≤a 2 ≤ ··· ≤an=mbe positive integers. Denote bybkthe number of
thoseaifor whichai≥k. Prove that

a 1 +a 2 +···+an=b 1 +b 2 +···+bm.

522.Show that fors>0,
∫∞

0

e−sxx−^1 sinxdx=arctan(s−^1 ).