1 Irrationality of rational series

Note di Matematica 29, n. 1, 2009, 73–77.

Irrationality for a class of rational series

Eduardo Pascali

Department of Mathematics ”‘Ennio De Giorgi”’, University of Salento, P.O.B.123, 73100, Lecce, Italy eduardo.pascali@unile.it

Received: 10/09/2007; accepted: 22/04/2008.

Abstract. The aim of this paper is to prove the irrationality of the sum of some classes of series of positive rational numbers.

Keywords:Irrationality, infinite series, infinite sequences MSC 2000 classification:primary 11J72

1 Irrationality of rational series

Various criteria for the irrationality of the sum of infinite series of positive rationals are known, see the references. We prove another criterion, which is strictly related to the proof in [6] and to the results in [3], [4], [8].

The first result is an easy step for the final criterion and clarify the role of the hypotheses.

1 Proposition. Given a

₁

> 1, a

₂

> 1 integers numbers, let (a

n

) be the sequence defined by

a

_n+1

= (a

₁

a

₂

· · · a

n

)

^γ1

, n > 2, with 1 < γ

₁

∈ N.

Let (b

n

) be a sequence of positive integers such that:

∃k

1

, γ

₂

∈]0, +∞[ b

n

≤ k

1

a

^γ_n−1²

∀n > 2.

If γ

₂

∈ [0, 1] and γ

1

(1 − γ

2

) ≥ 1, then the series 

j bj

aj

is convergent to an irrational number. Moreover for every non decreasing sequence (c

_j

) of positive integers the series 

j bj

ajcj

is convergent to an irrational number.

Proof. We remark that (a

_n

) is an increasing sequence and that by induc- tion it can be checked that a

_n+2

≥ 2

^(2γ1)n

for all n. Moreover, the following inequalities hold:

0 < b

j

a

_j

≤ k

1

(a

_j−1

)

^γ2

a

_j

= k

₁

1

(a

₁

. . . a

_j−2

)

^γ1(1−γ2)

(a

_j−1

)

^γ1

≤k

1

1

a

^γ_j−1¹

≤ k

1

1

2

^{γ1(2γ1)j−3}

. Note Mat. 29 (2009), n. 1, 73-78

ISSN 1123-2536, e-ISSN 1590-0932 DOI 10.1285/i15900932v29n1p73

http://siba-ese.unisalento.it, © 2009 Università del Salento

__________________________________________________________________

As a consequence, the series 

j bj

aj

is convergent. Arguing as in [6], assume that the sum is a rational number α =

^p_q

, where p, q ∈ N and 1 ≤ q, and observe that:

A

n

= &

p − q



n j=1

b

j

a

_j

) *

ⁿ

j=1

a

j

= q & *

ⁿ

j=1

a

j

) 

^+∞

j=n+1

b

j

a

_j

is a positive integer and A

n

≥ 1. Now:

A

_n

=q & *

ⁿ

j=1

a

_j

) 

^+∞

j=n+1

b

_j

a

j

= qa

γ11 n+1



+∞

j=n+1

b

_j

a

j

= q

+∞



j=n+1

b

j

a

γ11 n+1

a

j

≤q

^+∞



j=n+1

b

j

a

γ11 j

a

j

≤ k

1

q

+∞



j=n+1

a

γ21 j−1

a

γ1−1γ1 j

≤ k

1

q



+∞

j=n+1

1

(a

₁

. . . a

_j−2

)

^{γ1−γ1γ2−1}

a

^γ_j−1¹⁻¹

≤k

1

q

+∞



j=n+1

1 [2

^γ1−1

]

^(2γ1)j−3

.

Then from the convergence of the series 

_+∞

j=n+1 1

[2^γ1−1]^(2γ1)j−3

, we have A

_n

< 1 for n large and a contradiction follows. The last statement easily follows from the inequalities:

b

j

c

j

a

j

≤ b

j

a

j

and, since (c

j

) is non a decreasing sequence:

+∞



j=n+1

b

_j

c

j

a

j

*

n j=1

(c

j

a

j

) ≤

+∞



j=n+1

c

_n+1

b

_j

c

j

a

j

*

n j=1

a

j

≤

+∞



j=n+1

b

_j

a

j

*

n j=1

a

j

.

QED

Now we can state our main result.

2 Theorem. Given a

₁

> 1, a

₂

> 1, and let (a

_n

) be a sequence of positive integers such that:

∃k

1

, k

₂

, α

₁

, α

₂

∈]0, +∞[ k

1

(a

₁

a

₂

· · · a

n

)

^α1

≤ a

n+1

≤ k

2

(a

₁

a

₂

· · · a

n

)

^α2

(1) for all n > 3. Let (b

_n

) be a sequence of positive integers such that

∃k

3

, γ ∈ [0, +∞[ b

n

≤ k

3

a

^γ_n−1

∀n > 3. (2) If the following conditions hold:

α

₁

− γα

2

≥ 0 (3)

(α

₁

+ α

₂

) α

₁

α

₂

> 1; k

α21 2

k

₁^α1

1 2

^{α21((α1+α2)α1α2−1)}

< 1 (4)

α

₁

> 1 (5)

k

₁

k

α1α2 2

≥ 1, (6)

then the series 

j bj

aj

converges to an irrational number; moreover if (c

j

) is an increasing sequences of positive integers then also 

j b_j

cjaj

converges to an irrational number.

Proof. By (1) and (6) we have:

a

_n+1

≥ k

₁

k

α1α2 2

a

^(α1+α2)

α1α2

n

≥ a

n

.

Hence the sequence (a

_n

) is non decreasing. Moreover from (1) we deduce:

a

_n+3

≥ k

_n

j=1(^α1_α2(α1+α2))^j 1

k

α1α2

_n−1

j=1(^α1_α2(α1+α2))^j 2

2

^{2α1((α1+α2)α1α2)n}

≡ D

n

and the following inequalities hold:

b

_n+4

a

n+4

≤ k

₃

a

^γ_n+3

a

n+4

≤ k

₃

k

^γ₂

k

1

1

(a

1

a

2

. . . a

n+2

)

^α1−γα2

a

^α_n+3¹

≤ k

₃

k

₂^γ

k

1

1 a

^α_n+3¹

≤ k

₃

k

γ

2

k

₁

D

⁻¹_n

. Consider now the series 

n

D

⁻¹_n

, and remark that:

D

⁻¹_n+1

D

⁻¹n

=

⎡

⎣k

α21 2

k

^α₁¹

1 2

^{2α1((α1+α2)α1α2)}

⎤

⎦

[^α1_α2(α1+α2]ⁿ

Then by (4), the series 

n

D

⁻¹_n

is convergent; hence also 

j bj

aj

is convergent.

Assume that the sum is a rational number

^p_q

, where p, q ∈ N and 1 ≤ q, as in the previous proposition. Observe that

A

n

= &

p − q



n j=1

b

j

a

j

) *

ⁿ

j=1

a

j

= q & *

ⁿ

j=1

a

j

)

^+∞



j=n+1

b

j

a

j

is a positive integer and A

n

≥ 1. But we remark that:

1 ≤A

n

≤ q k

α11 1

+∞



j=n+1

b

_j

a

α11 j

a

j

≤ k

₃

q k

α11 1

+∞



j=n+1

a

^γ_j−1

a

α1−1α1 j

≤ k

₃

q k

α11 1



+∞

j=n+1

k

₂^γ

(a

₁

. . . a

_j−2

)

^γα2

k

α1−1α1

1

(a

₁

. . . a

_j−1

)

^α1−1

≤ qk

₃

k

^γ₂

k

₁



+∞

j=n+1

1

(a

₁

. . . a

_j−2

)

^α1−α2γ

a

^α_j−1¹⁻¹

≤ qk

₃

k

₂^γ

k

₁

+∞



j=n+1

1 a

^α_j−1¹⁻¹

≤ qk

₃

k

^γ₂

k

₁



+∞

j=n+1

[ k

α1α2

_j−5

i=0(^α1_α2(α1+α2))ⁱ 2

k

_j−4

i=0(^α1_α2(α1+α2))ⁱ 1

1 2

^{2α1(α1α2(α1+α2))j−4}

]

^α1−1

= qk

₃

k

^γ₂

k

₁



+∞

j=n+1

[ k

α21

_j−5

i=0(^α1_α2(α1+α2))ⁱ 2

k

^α1

j−4

i=0(^α1_α2(α1+α2))ⁱ 1

1 2

^{2α12(α1α2(α1+α2))j−4}

]

^α1−1α1

= qk

₃

k

^γ₂

k

₁



+∞

j=n+1

(D

⁻¹_j−4

)

^α1−1α1

.

Since the series 

j

(D

_j⁻¹

)

^α1−1α1

is convergent, we have lim

_n

A

_n

= 0, a contradic- tion follows and the irrationality of the sum is proved.

The last statement can be proved as in Proposition 1.

^QED

Let us show some examples to which our result can be applied. If we choose α

₁

= 2 and α

₂

= 4, we can select 0 < γ ≤

¹₂

and 0 < k

₂

≤ k

1²

.

Theorem 2 ensures the following abstract scheme for the irrationality of the sum of a rational series.

3 Theorem. Consider a pair ((a

n

), ψ), where (a

n

) is a strictly increasing sequence of positive integers and ψ : [0, +∞[→]0, +∞[ is an increasing function.

Assume that

*

n j=1

a

j

≤ ψ(a

n+1

). (7)

and take h : [1, + ∞[→ [0, +∞[ non increasing such that



_+∞

1

h(ξ)dξ < +∞. (8)

Then for every sequence (b

j

) of positive integers for which:

b

_j

a

j

ψ(a

j

) ≤ h(a

j

), (9)

the series 

j bj

aj

is convergent and the sum is an irrational number.

Proof. Since b

j

a

j

≤ b

j

a

j

ψ(a

_j

) 1

ψ(a

j

) ≤ h(a

j

)

ψ(a

j

) ≤ h(a

j

) ψ(1) , for n > a

₁

we have



n j=1

b

_j

a

j

≤



n j=1

h(a

_j

) ψ(1) ≤ 1

ψ(1)



n j=1

h(a

_j

)(a

_j+1

− a

j

) a

_j+1

− a

j

≤ 1 ψ(1)



n j=1

h(a

j

)(a

_j+1

− a

j

) ≤ 1 ψ(1)



n a1

h(ξ)dξ.

Then the series 

j bj

aj

is convergent. Arguing as in the proof of Theorem 2, we have:

1 ≤ A

n

=q(

*

n j=1

a

j

)

+∞



j=n+1

b

j

a

j

≤ qψ(a

n+1

)

+∞



j=n+1

b

j

a

j

≤ q 

^+∞

j=n+1

b

j

ψ(a

j

) a

j

≤



+∞

j=n+1

h(a

j

) ≤



+∞

j=n+1

h(a

j

)(a

_j+1

− a

j

) ≤ q



_+∞

an

h(ξ)dξ.

From (8), A

n

→ 0 and we get a contradiction.

^QED

References

[1] Badea C.: The irrationality of certain infinite series; Glasgow Math. J. 29 (1987), 221–

228.

[2] Badea C.: A theorem on irrationality of infinite series and applications; Acta Arith. XIII, 4 (1993), 313–323.

[3] Erd¨os P.: Some problems and results on the irrationality of the sum of infinite series; J.

Math. Sci., 10 (1975), 1–7.

[4] Erd¨os P. and Strauss E.G.: On the irrationality of certain Ahmes series; J. Indian Math. Soc., 27 (1963), 129–133.

[5] Hanˇcl J.: Criterion for irrational sequences; J. Number Theory, no. 1 43 (1993), 88–92.

[6] Hanˇcl J.: Irrational rapidly convergent series; Rend. Sem. Mat. Univ. Padova 107 (2002), 225–231.

[7] Hanˇcl J.: Irrational sequences of rational numbers; Note di Mat. (2004), 1–10.

[8] S´andor J.: Some classes of irrational numbers; Studia Univ. Babes-Bolyai, Math. XXIX (1984), 3–12.