Advanced book on Mathematics Olympiad

160 3 Real Analysis

f(x)=f(a)+

f′(a)

1!

(x−a)+

f′′(a)

2!

(x−a)^2 +···+

f(n)(a)

n!

(x−a)n+···.

Ifa =0, the expansion is also known as the Maclaurin series. Rational functions,
trigonometric functions, the exponential and the natural logarithm are examples of ana-
lytic functions. A particular example of a Taylor series expansion is Newton’s binomial
formula

(x+ 1 )a=

∑∞

n= 0

(

a

n

)

xn=

∑∞

n= 0

a(a− 1 )···(a−n+ 1 )

n!

xn,

which holds true for all real numbersaand for|x| <1. Here we make the usual
convention that

(a

0

)

=1.

We begin our series of examples with a widely circulated problem.

Example.Compute the integral

∫ 1

0

lnxln( 1 −x)dx.

Solution.Because

lim

x→ 0

lnxln( 1 −x)=lim

x→ 1

lnxln( 1 −x)= 0 ,

this is, in fact, a definite integral.
We will expand one of the logarithms in Taylor series. Recall the Taylor series
expansion

ln( 1 −x)=−

∑∞

n= 1

xn

n

, forx∈(− 1 , 1 ).

It follows that on the interval( 0 , 1 ), the antiderivative of the functionf(x)=lnxln( 1 −
x)is

∫

ln( 1 −x)lnxdx=−

∫ ∑∞

n= 1

xn

n

lnxdx=−

∑∞

n= 1

1

n

∫

xnlnxdx.

Integrating by parts, we find this is to be equal to

−

∑∞

n= 1

1

n

(

xn+^1

n+ 1

lnx−

xn+^1

(n+ 1 )^2

)

+C.

Taking the definite integral over an interval[, 1 −], then letting→0, we obtain