Irrationality of quick convergent series

J

AROSLAV

H

AN ˇ

CL

Irrationality of quick convergent series

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{2 (1996),}

p. 275-282

<http://www.numdam.org/item?id=JTNB_1996__8_2_275_0>

© Université Bordeaux 1, 1996, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Irrationality

of

_quick

_convergent

series

par JAROSLAV

HAN010CL

RÉSUMÉ. On démontre une _{généralisation} d’un résultat dû à Badea

concer-nant l’irrationnalité de certaines séries à convergence rapide.

ABSTRACT. We generalize a previous result due to Badea _relating to the irrationality of some _{quick convergent} infinite series.

There are _{many papers}

_concerning

the

irrationality

of infinite series.

Erdos

_[4]

_proved

that if the sequence of

positive integers

con-00

verges

quickly

to

_infinity,

then the series

_E

is an irrational number. ~=1

The author

_[7]

defined the irrational sequences and

proved

criterion for them. Another result is due to Erdos and Strauss

_{[5] .}

_They

_proved

that if

is a _sequence of

_{positive integers}

with o0

n 00

and

1,

then the

number £

_1fan

is rational if and

_only

n=1

if _{an+i =}

_{an -- an}

+ 1 holds for _{every n} &#x3E; no . Sander

[8]

_proved

that

if and are two sequences of

positive

integers

such that

= _oo and _&#x3E;

_1,

then the 00

number

_E

_bnfan

is irrational. n=1

Finally

Badea

_[1]

proved

that if and

_{{bn }°°_ 1}

are two _sequences

of

_{positive integers}

such that &#x3E;

bn)an+I/an

+

_1,

then the sum

00

E

is an irrational number. Later he

_generalized

his result

_{( [2] ) .}

n=i

In this _paper we will

_generalize

Badea’s result in another _{way and prove} t he

_{fol lowing}

theorem.

THEOREM. Let sequences

of positive

integers.

If there is a

natural number rri such that the

following

three

inequalities

hold

for

every n &#x3E; no

00

then the number A

_{= ~}

irrational.

~=1

Proo f :

For the sake of

simplicity

we will suppose that

(1) - (3)

hold for every n.

(If

_not, we define

a~

= = for _{every n} =

1,2,...

and these two _sequences and

_satisfy

then our above

_{requirements.)}

Let us denote

One can _prove

_by

induction that

and

hold for i =

0,1,... ,

m + 1.

for _every natural number n. Then

_{(4) - (9~}

imply

that &#x3E; and

_An,i

&#x3E; 0

for every

positive integer n

and

_z=0~1~"

m -1.

First we will _prove that

and

_secondly

(11)

implies

that there is a number c &#x3E; 0 such that

Using

the famous theorem of Stolz

_(see

_e.g.

_(6~),

we obtain

On the other hand

_(10),

_{(11), (12)}

and Brun’s Theorem

_(see

_e.g.

_(3~)

_imply

the

_{irrationality}

of the number A.

(13)

and

₍₂₎

_yield

_(10).

_Similarly

_{(13) (if}

substitute + 1 instead of

m)

and

₍₃₎

_yield

_(11).

Remark: If we

_put

m = 0 in the main

theorem,

then we receive

bn

&#x3E;

1,

0 &#x3E; -an and

1) -

Thus

bn

&#x3E;

+1 and this is the famous theorem due to Badea

_(see

e.g.

[1]).

Consequence

1: Let and be two _sequences of

_positive

integers.

If

00

hold for every n &#x3E; _no, then the number A

_{= £}

is irrational. n=1

Proof:

Let us _put m = 1 in the main theorem. Then

(14)

is

(1), (15)

is

(2)

and

₍₁₆₎

is

_(3).

Consequence

2: Let be a _sequence of

_{positive integers}

such that

bl

&#x3E; 2 and

hold for every n &#x3E; no where k is a

_{positive integer.}

Then the number

00

A

_{= L}

is irrational.

n=1

Let us

_put

an in consequence 1. Then

(17)

immediately implies

(15) and

This and

_bl

&#x3E; 2

_imply

that the sequence is

increasing.

Thus

₍₁₅₎

Let us define the _sequence of

_nonnegative

_integers

such that

(17)

implies

that

Substituting

(19)

for

₍₁₈₎

we obtain the

_{equivalent inequality}

₍₂₁₎

with

Carrying

out the

_equivalent

calculations

_step

_by

step,

we receive

Using

(20)

and the fact that &#x3E;

₂₎

is an

_increasing

_{sequence, it}

is

_enough

to prove that

where K is a suitable constant.

₍₂₂₎

is

_equivalent

with

(17)

implies

that

Because of

_(23),

it is

_enough

to prove that

where

_K,

is a suitable constant too. But

₍₂₄₎

is true _{for every n &#x3E; no.}

Thus

₍₁₈₎

is

_right

and the number A is irrational.

REFERENCES

[1]

C. Badea, The Irrationality of Certain _{Infinite Series, Glasgow.} Math J. 29

_(1987),

221-228.

[2]

C. _Badea, A Theorem on _{Irrationality of Infinite} Series and Applications, Acta Arith. LXII.4

_(1993),

313-323.

[3]

V. _Brun, A Theorem about Irrationality, Arch. for Math. _og Naturvideskab Kris-tiana _{31, ,}

_(1910),

3

_(German).

[4]

P. Erdös, Some Problems and Results on the Irrationality of the Sum of Infinite Series, J. Math. Sci. 10

_{(1975), 1-7.}

[5]

P. Erdös and E. G. Strauss, On the Irrationality of Certain Ahmes _Series, J. Indian Math. Soc. 27

_(1963),

129-133.

[6]

G. M. _Fichtengolz, The Lecture of Differential and _{Integral Calculations, part I,}

issue _{6, Nauka, Moskva, 1966}

_(Russian).

[7]

J. Han010Dl, Criterion _for Irrational Sequences, J. Num. _Theory 43 n° 1 _(1993), 88-92.

[8]

J. Sándor, Some Classes of Irrational Numbers, Studia Univ. _Babes-Bolyai Math.

29 _(1984), 3-12. Jaroslav HANCL Department of Mathematics University of Ostrava Dvorakova 7 701 03 Ostrava 1 Czech _Republic