Advanced book on Mathematics Olympiad

3.1 Sequences and Series 119

Clearly, the left-hand side of this equality is an integer. For the right-hand side, we have

0 <

∑

j>k

qn 1 n 2 ···nk

nj

≤

qn 1 n 2 ···nk

nk+ 1

+

qn 1 n 2 ···nk

3 nk+ 1

+··· ≤

1

3

+

1

9

+

1

27

+··· =

1

2

.

Here we used the fact thatn 1 n 2 ···nkn+k 1 ≤ 31 qand thatnj+ 1 ≥ 3 nj, forj≥kandk
sufficiently large. This shows that the right-hand side cannot be an integer, a contradiction.
It follows that the sum of the series is irrational. 

352.Show that the series

1

1 +x

+

2

1 +x^2

+

4

1 +x^4

+···+

2 n

1 +x^2 n

+···

converges when|x|>1, and in this case find its sum.

353.For what positivexdoes the series

(x− 1 )+(

√

x− 1 )+(^3

√

x− 1 )+···+(n

√

x− 1 )+···

converge?

354.Leta 1 ,a 2 ,...,an,...be nonnegative numbers. Prove that

∑∞

∑∞ n=^1 an<∞implies

n= 1

√

an+ 1 an<∞.

355.LetS={x 1 ,x 2 ,...,xn,...}be the set of all positive integers that do not contain
the digit 9 in their decimal representation. Prove that

∑∞

n= 1

1

xn

< 80.

356.Suppose that(xn)nis a sequence of real numbers satisfying

xn+ 1 ≤xn+

1

n^2

, for alln≥ 1.

Prove that limn→∞xnexists.

357.Does the series

∑∞

n= 1 sinπ

√

n^2 +1 converge?

358.(a) Does there exist a pair of divergent series

∑∞

n= 1 an,

∑∞

n= 1 bn, witha^1 ≥a^2 ≥

a 3 ≥ ··· ≥0 andb 1 ≥b 2 ≥b 3 ≥ ··· ≥0, such that the series

∑

nmin(an,bn)is

convergent?

(b) Does the answer to this question change if we assume additionally thatbn=n^1 ,

n= 1 , 2 ,...?