Advanced book on Mathematics Olympiad

3.1 Sequences and Series 109

Weierstrass’ theorem.A monotonic bounded sequence of real numbers is convergent.

Below are some instances in which this theorem is used.

323.Prove that the sequence(an)n≥ 1 defined by

an= 1 +

1

2

+

1

3

+···+

1

n

−ln(n+ 1 ), n≥ 1 ,

is convergent.

324.Prove that the sequence

an=

√

1 +

√

2 +

√

3 +···+

√

n, n≥ 1 ,

is convergent.

325.Let√ (an)nbe a sequence of real numbers that satisfies the recurrence relationan+ 1 =
an^2 +an−1, forn≥1. Prove thata 1 ∈/(− 2 , 1 ).

326.Using the Weierstrass theorem, prove that any bounded sequence of real numbers
has a convergent subsequence.

Widely used in higher mathematics is the following convergence test.

Cauchy’s criterion for convergence.A sequence(xn)nof points inRn(or, in general,
in a complete metric space)is convergent if and only if for any> 0 there is a positive
integernsuch that whenevern, m≥n,‖xn−xm‖<.

A sequence satisfying this property is called Cauchy, and it is the completeness of
the space (the fact that it has no gaps) that forces a Cauchy sequence to be convergent.
This property is what essentially distinguishes the set of real numbers from the rationals.
In fact, the set of real numbers can be defined as the set of Cauchy sequences of rational
numbers, with two such sequences identified if the sequence formed from alternating
numbers of the two sequences is also Cauchy.

327.Let(an)n≥ 1 be a decreasing sequence of positive numbers converging to 0. Prove
that the seriesS=a 1 −a 2 +a 3 −a 4 +···is convergent.

328.Leta 0 ,b 0 ,c 0 be real numbers. Define the sequences(an)n,(bn)n,(cn)nrecur-
sively by

an+ 1 =

an+bn

2

,bn+ 1 =

bn+cn

2

,cn+ 1 =

cn+an

2

,n≥ 0.

Prove that the sequences are convergent and find their limits.