Advanced book on Mathematics Olympiad

120 3 Real Analysis

359.Given a sequence(xn)nwithx 1 ∈( 0 , 1 )andxn+ 1 =xn−nxn^2 forn≥1, prove that
the series

∑∞

n= 1 xnis convergent.

360.Is the number

∑∞

n= 1

1

2 n^2

rational?

361.Let(an)n≥ 0 be a strictly decreasing sequence of positive numbers, and letzbe a
complex number of absolute value less than 1. Prove that the sum

a 0 +a 1 z+a 2 z^2 +···+anzn+···

is not equal to zero.

362.Letwbe an irrational number with 0 <w<1. Prove thatwhas a unique
convergent expansion of the form

w=

1

p 0

−

1

p 0 p 1

+

1

p 0 p 1 p 2

−

1

p 0 p 1 p 2 p 3

+···,

wherep 0 ,p 1 ,p 2 ,...are integers 1≤p 0 <p 1 <p 2 <···.

363.The numberqranges over all possible powers with both the base and the exponent
positive integers greater than 1, assuming each such value only once. Prove that
∑

q

1

q− 1

= 1.

364.Prove that for anyn≥2,

∑

p≤n,pprime

1

p

>ln lnn− 1.

Conclude that the sum of the reciprocals of all prime numbers is infinite.

3.1.6 Telescopic Series and Products.............................

We mentioned earlier the idea of translating notions from differential and integral calculus
to sequences. For example, the derivative of(xn)nis the sequence whose terms are
xn+ 1 −xn,n≥ 1, while the definite integral is the sumx 1 +x 2 +x 3 + ···. The
Leibniz–Newton theorem