Irrational rapidly convergent series

Irrational Rapidly Convergent Series.

JAROSLAV HANCˇ_L(*)

ABSTRACT- The main result of this paper is a criterion for irrational series which consist of rational numbers and converge very quickly.

1. Introduction.

Mahler in [6] introduced the main method of proving the irrationality of sums of infinite series. This method has been extended several times and Nishioka’s book [7] contains a survey of these results. Other methods are given in Sándor [8], Hancˇl [5] and Erdös [4].

In 1987 in [1] Badea proved the following theorem.

THEOREM 1. Let]a_n(^n41Q and ]b_n(^n41Q be two sequences of positive integers such that for every large n, an11D^bn11_b

n

an2

2^bn11_b

n

an11. Then the sum a4

!

k41 Q bn

a_n is an irrational number.

Later in [2] he improved this result. Erdös in [4] introduced the no- tion of irrational sequences of positive integers and proved that the se- quence ]2²ⁿ(n41Q is irrational. In [5] the present author extended this definition of irrational sequences to sequences of positive real num- bers.

(*) Indirizzo dell’A.: Department of Mathematics, University of Ostrava, Dvorˇákova 7, 701 03 Ostrava 1, Czech Republic, e-mail: hanclHosu.cz

Supported by the grant 201/01/0471 of the Czech Grant Agency.

DEFINITION 1. Let]an(ⁿQ41be a sequence of positive real numbers.

If, for every sequence]c_n(ⁿQ41of positive integers, the sum

!

n41 Q 1

a_nc_n is an irrational number then the sequence]a_n(n4Q 1is called irrational. If the sequence is not irrational then it is rational.

The same paper also gives a criterion for the irrationality of infinite sequences and series.

In 1999 in [3] Duverney proved the following theorem.

THEOREM 2. Let b be a positive rational number and g be a non- negative real number with 0GgE2. Assume that ]u_n(ⁿQ41, ]a_n(ⁿQ41

and ]bn(ⁿQ41 are sequences of nonzero integers such that

nlimK Qu_n4 Q, un114bun2

1O(ung), logNa_nN 4o( 2ⁿ) and

logNbnN 4o( 2ⁿ).

Thena4

!

n41 Q a_n

b_nu_n is a rational number if and only if there is n0such that for every nDn₀

u_n114bu_n²2a_n11b_n

a_nb_n11u_n1 a_n12b_n11 ba_n11b_n12.

The main result of this paper is Theorem 3. It deals with a criterion for the irrationality of sums of infinite series of rational numbers which depends on the speed and character of the convergence. In particular it does not depend on arithmetical properties like divisibility.

THEOREM 3. Let AD1be a real number. Suppose that]d_n(ⁿQ41is a sequence of real numbers greater than one. Let]an(ⁿQ41and ]bn(ⁿQ41be two sequences of positive integers such that

nlimK Qa_n

1 2n

4A (1)

and for all sufficiently large n A a_n

1 2n

D

»

j4n Q

dj

(2)

and

nlimK Q

dn2ⁿ

bn

4 Q. (3)

Then the series a4

!

n41 Q b_n

a_n is an irrational number.

COROLLARY 1. Let AD1 be a real number. Assume that ]a_n(^n41Q and]bn(Qⁿ41 are two sequences of positive integers such that lim

nKQa_n

1 2n

4A and for every sufficiently large positive integer n, a_n

1

2n

g

h

bnG2

1 n42n

. Then the series

!

n41 Q b_n

a_n is irrational.

This is an immediate consequence of Theorem 3. It is enough to put d_n411_n¹3.

COROLLARY 2. Let AD1 be a real number. Assume that ]a_n(^n41Q and]bn(Qⁿ41 are two sequences of positive integers such that lim

nKQa_n

1 2n

4A and for every sufficiently large positive integer n, a_n

1

2n (114( 2 /3 )ⁿ)_GA and b_nG2^{( 4 /3 )n21}. Then the series

!

n41 Q b_n

a_n is irrational.

This is an immediate consequence of Theorem 3. It is enough to put d_n411( 2 /3 )ⁿ.

REMARK 1. Theorem3does not hold if we omit condition(2).To see this let a₀be a positive integer greater than 1and for every positive in- teger n, a_n4a_n²₂₁2a_n2111. Then

a_n

1 2n11

4a_n

1

2n

g

an221

h

and

a_n

1 2n11

F

g

h

g

h

nK Qa_n

1 2n

D1 and

n4

!

a_n 4 1 a₀ 1

!

n41 Q 1

a_n4

4 1 a₀1_n

!

41

Q a_n21 a_n(a_n21 ) 4 1

a₀1_n

!

41

Q

g

a_n(a_n21 )

h

4 1 a0

1

!

n41

Q

g

»

(a₀21)

»

h

1 1

a0(a021)4 1 a021 is a rational number.

EXAMPLE 1. Let[x] be the greatest integer less then or equal to x, p(n) be the number of primes less then or equal to n and d(n) be the number of positive divisors of the number n. As an immediate conse- quence of Corollary 1 we obtain that the series

n

!

Q [d(n)^{( 3 /2 )n}]12n

y g

h

^z

^!

Q [d(n)^{2( 4 /3 )n}]1n²

y g

h

^z

are irrational numbers.

EXAMPLE 2. Let[x],p(n)and d(n)be defined as in Example1.As an immediate consequence of Corollary 2 we obtain that the series

n4

!

Q [d(n)^{( 5 /4 )n}]1n

kg

h

l

^!

Q [d(n)^{( 6 /5 )n}]1n²

kg

h

l

are irrational numbers.

OPEN PROBLEM 1. It is an open problem if for every sequence ]c_n(ⁿQ41 of positive integers the sum

!

n41

Q 1

c_n2²ⁿ12ⁿ12 is irrational or not.

2. Proof.

PROOF(of Theorem 3). From (1) and (2) we obtain that lim

nK Qa

n 1 2n11

4 4kAand lim

nK Qd_n41. This, (1) and (3) imply lim

nK Q(d_n11a

n 1 2n11a

n11 2 ¹

2n11)4 4 ¹

kAE1. From this and (3) we obtain that

(4) lim

nK Q

b_n11 a_n11 b_n an

4 lim

nK Q

g

b_n

h

nK Q(d_n11a

n 1 2n11a

n11 2 ¹

2n11)²ⁿ¹¹40 . Inequality (4) implies that the series

!

n41 Q b_n

a_n is convergent and for every sufficiently large positive integer n

j4n11

!

Q b_j aj

G2b_n11 an11

. (5)

Let us suppose that the series a4

!

n41 Q b_n

an

is a rational number. Then there exist positive integers p and q such that a4^p_q. Thus we have

a4 p q 4

!

j41 Q b_j

a_j4

!

j41 n b_j

aj

1

!

j4n11 Q b_j

aj

. It implies that

A_n4

u

^!

v

^»

^»

^!

(6)

is a positive integer for every n41 , 2 ,R. Thus A_nF1 .

(7)

Now we find a positive integer nsuch that AnE1, a contradiction with (7). Lets be a positive integer such that for every nFs, (2) holds. This and (2) imply that for every n and k with sGkGn

A ak

1 2k

D

»

j4k Q

d_jF

»

j4n Q

d_j.

Hence

A

j4maxs,R,na_j

1 2j

D

»

j4n Q

d_j. (8)

From (1) and (8) we see that for infinitely many n an11

1 2n11

Fd_n11 max

j4s,R,na

j 1 2j

(9)

otherwise there exists n₀ with n₀Fs such that for every nFn₀ an11

1 2n11

Edn11 max

j4s,R,na_j

1 2j

GRG

»

j4n011 n11

dj max

j4s,R,n0

a_j

1 2j,

contradicting (1) and (8) for sufficiently large n. Inequality (9) and the fact that 2^s1R12ⁿE2ⁿ¹¹ imply

an11Fdn21ⁿ¹¹1

g

h

g

h

4d_n11²ⁿ¹¹

»

i4s

n

g

1

2j

h

»

j4s n

a_j4d_n²₁ⁿ¹¹₁

g

^»

h g

^»

a_j

h

From this, (3), (5) and (6) we obtain for infinitely many n A_n4q

g

^»

n

a_j

h

^!

a_j Gq

g

^»

a_j

h

E2q

g

^»

aj

h

n112ⁿ¹¹(

»

»

g

^»

a_j

h

n21ⁿ¹1¹ E1 and the proof of Theorem 3 is complete. r

Acknowledgement. I would like to thank Terence Jackson for his comments on an earlier draft of this paper.

R E F E R E N C E S

[1] C. BADEA, The irrationality of certain infinite series, Glasgow Math. J.,29 (1987), pp. 221-228.

[2] C. BADEA,A theorem on irrationality of infinite series and applications, Ac- ta Arith., LXIII (1993), pp. 313-323.

[3] D. DUVERNEY,Sur les series de nombres rationnels a convergence rapide, C.

R. Acad. Sci., ser.I, Math., 328, no. 7 (1999), pp. 553-556.

[4] P. ERDO¨_S,Some problems and results on the irrationality of the sum of infi- nite series, J. Math. Sci., 10 (1975), pp. 1-7.

[5] J. HANCˇ_LJ.,Criterion for irrational sequences, J. Number Theory,43, no. 1 (1993), pp. 88-92.

[6] K. MAHLER,Lectures on Transcendental numbers, Lecture notes in mathe- matics 546, Springer, 1976.

[7] K. NISHIOKA,Mahler Functions and Transcendence, Lecture notes in mathe- matics 1631, Springer, 1996.

[8] J. SA´_NDOR,Some classes of irrational numbers, Studia Univ. Babes Bolyai, Mathematica,29 (1984), pp. 3-12.

Manoscritto pervenuto in redazione il 25 ottobre 2001.