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REND. SEM. MAT. UNIV. PADOVA, Vol. 107 (2002)
Irrational Rapidly Convergent Series.
JAROSLAV HANČL (*)
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], Hančl [5] and Erdös [4].
In 1987 in [1] Badea proved the following theorem.
THEOREM 1.
Let ]an (nQ4 1 and ]bn (nQ4 1 be two sequences of positive
integers such that for every large n , an 1 1 D
the sum a 4
Q
bn
k41
an
!
bn 1 1
bn
an2 2
bn 1 1
bn
an 1 1. Then
is an irrational number.
Later in [2] he improved this result. Erdös in [4] introduced the notion of irrational sequences of positive integers and proved that the sen
quence ] 22 (nQ4 1 is irrational. In [5] the present author extended this
definition of irrational sequences to sequences of positive real numbers.
(*) Indirizzo dell’A.: Department of Mathematics, University of Ostrava,
Dvořákova 7, 701 03 Ostrava 1, Czech Republic, e-mail: hanclHosu.cz
Supported by the grant 201/01/0471 of the Czech Grant Agency.
226
Jaroslav Hančl
DEFINITION 1.
Let ]an (nQ4 1 be a sequence of positive real numbers.
If, for every sequence ]cn (nQ4 1 of positive integers, the sum
Q
1
n41
an cn
!
is an
irrational number then the sequence ]an (nQ4 1 is 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 nonnegative real number with 0 G g E 2. Assume that ]un (nQ4 1 , ]an (nQ4 1
and ]bn (nQ4 1 are sequences of nonzero integers such that
lim un 4 Q ,
nKQ
un 1 1 4 bun2 1 O(ung ),
log Nan N 4 o( 2n )
and
log Nbn N 4 o( 2n ).
Then a 4
Q
an
n41
bn un
!
is a rational number if and only if there is n0 such
that for every n D n0
un 1 1 4 bun2 2
a n 1 1 bn
an bn 1 1
un 1
an 1 2 bn 1 1
ban 1 1 bn 1 2
.
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 A D 1 be a real number. Suppose that ]dn (nQ4 1 is a
sequence of real numbers greater than one. Let ]an (nQ4 1 and ]bn (nQ4 1 be
two sequences of positive integers such that
1
(1)
lim an2n 4 A
nKQ
Irrational rapidly convergent series
227
and for all sufficiently large n
A
(2)
1
Q
D
an2n
»d
j4n
j
and
n
(3)
lim
nKQ
Q
Then the series a 4
COROLLARY 1.
!
bn
dn2
bn
4Q.
is an irrational number.
n 4 1 an
Let A D 1 be a real number. Assume that ]an (nQ4 1
1
2n
and ]bn (Q
n41 are two sequences of positive integers such that lim an 4A
g
1
and for every sufficiently large positive integer n, an2n 1 1
bn G 2
1 n
2
n4
Q
bn
n41
an
!
. Then the series
nKQ
1
n
h G A and
is irrational.
This is an immediate consequence of Theorem 3. It is enough to put
dn 4 1 1
1
n3
.
COROLLARY 2.
Let A D 1 be a real number. Assume that ]an (nQ4 1
1
2n
and ]bn (Q
n41 are two sequences of positive integers such that lim an 4A
nKQ
1
and for every sufficiently large positive integer n, an2n (114( 2 /3 )n )GA
and bn G 2( 4 /3 )
Q
bn
n41
an
!
n21
. Then the series
is irrational.
This is an immediate consequence of Theorem 3. It is enough to put
dn 4 1 1 ( 2 /3 )n.
REMARK 1. Theorem 3 does not hold if we omit condition (2). To see
this let a0 be a positive integer greater than 1 and for every positive integer n , an 4 an22 1 2 an 2 1 1 1. Then
1
2n 1 1
an
1
2n
4 an
g
12
1
an 2 1
1
1
an22 1
h
1
2n 1 1
1
G an2n
228
Jaroslav Hančl
and
1
2n 1 1
an
F
gh
3
1
2n 1 1
4
1
an F
2n
gh
3
12
1
2n
4
a0 F
3
5
a0 D 1 .
1
Hence lim an2n D 1 and
nKQ
Q
Q
! a1 4 a1 1 ! a1 4
n40
4
4
1
a0
a0
n
! a a(a2211 ) 4 a1 1 ! g a 21 1 2 a (a12 1 ) h 4
Q
1
1
n41
0
n
Q
n
n41
n
0
n
! g (a 21)1»
Q
1
n41
n21
j40 aj
0
2
n41
n
n
1
(a021)
»
n
j40 aj
h
4
1
a0
n
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 consequence of Corollary 1 we obtain that the series
n
[d(n)( 3 /2 ) ] 1 2 n
Q
!
n42
ygk2 2 p(n) h z 2 n!
2n
1
n
[d(n)2( 4 /3 ) ] 1 n 2
Q
and
!
n42
1
ygk2 2 p(n)
h z2n
2n
n
are irrational numbers.
EXAMPLE 2. Let [x], p(n) and d(n) be defined as in Example 1. As
an immediate consequence of Corollary 2 we obtain that the series
n
[d(n)( 5 /4 ) ] 1 n
Q
!
n42
kg
k2 2
hl
2n
1
2p(n)
n
Q
!
and
2n n
n42
kg
[d(n)( 6 /5 ) ] 1 n 2
k2 2
1
2p(n)
hl
2n
2 n!
are irrational numbers.
OPEN
PROBLEM
1.
It is an open problem if for every sequence
]cn (nQ4 1 of positive integers the sum
not.
Q
!
n41
1
2n
cn 2 1 2 n 1 2
is irrational or
Irrational rapidly convergent series
229
2. Proof.
1
PROOF (of Theorem 3). From (1) and (2) we obtain that lim an2n 1 1 4
1
4 kA and lim dn 4 1. This, (1) and (3) imply lim (dn 1 1 an
4
1
kA
nKQ
nKQ
1
an 121n 1 1 ) 4
E 1. From this and (3) we obtain that
bn 1 1
(4)
nKQ
2
2n 1 1
lim
nKQ
an 1 1
bn
4 lim
nKQ
g
h
bn 1 1 an
an 1 1 bn
an
n11
G lim dn21 1
nKQ
an 1 1
4
an
1
2
1
4 lim (dn 1 1 an2n 1 1 an 121n 1 1 )2
nKQ
Q
bn
n41
an
!
Inequality (4) implies that the series
sufficiently large positive integer n
bj
Q
!
(5)
aj
j4n11
Let us suppose that the series a 4
an 1 1
Q
bn
n41
an
!
.
is a rational number. Then
there exist positive integers p and q such that a 4
have
a4
p
q
n
Q
bj
j4n11
aj
! ab 4 ! ab 1 !
Q
4
j
j41
j
j41
j
40.
is convergent and for every
2 bn 1 1
G
n11
j
p
q
. Thus we
.
It implies that
(6)
u
An 4 p 2 q
! ab v » a 4 q » a !
n
n
j
j41
n
j
j41
j
j41
j
Q
bj
j4n11
aj
is a positive integer for every n 4 1 , 2 , R . Thus
An F 1 .
(7)
Now we find a positive integer n such that An E 1, a contradiction with
(7). Let s be a positive integer such that for every n F s, (2) holds. This
and (2) imply that for every n and k with s G k G n
A
1
ak2k
Q
D
Q
»d F »d .
j4k
j
j4n
j
230
Jaroslav Hančl
Hence
Q
A
(8)
D
1
max aj 2j
»d.
j
j4n
j4s, R, n
From (1) and (8) we see that for infinitely many n
1
1
n11
an21
F dn 1 1 max aj 2j
1
(9)
j4s, R, n
otherwise there exists n0 with n0 F s such that for every n F n0
1
n11
1
1
»
n11
j
an21
1 E dn 1 1 max aj 2 G R G
j4s, R, n
j 4 n0 1 1
dj max aj 2j ,
j 4 s , R , n0
contradicting (1) and (8) for sufficiently large n. Inequality (9) and the
fact that 2s 1 R 1 2n E 2n 1 1 imply
n11
an 1 1 F dn21 1
g
1
max aj 2j
j4s, R, n
2n 1 1
n
n11
4 dn21 1
h
»g
n11
D dn21 1
1
max aj 2j
i4s j4s, R, n
h
2i
g
1
max aj 2j
j4s, R, n
2n 1 2 n 2 1 1 R 1 2 s
4
» a 4d g » a h g » a h
n
n11
F dn21 1
h
j
j4s
2n 1 1
n11
s21
n
j41
j
j41
21
j
From this, (3), (5) and (6) we obtain for infinitely many n
g» h !
g» h
n
An 4 q
j41
Q
aj
j4n11
bj
aj
n
E2q
j41
aj
g» h
n
Gq
j41
2 bn 1 1
aj
an 1 1
E
dn21 1
(
»
n
j 4 1 aj )(
g» h
s21
bn 1 1
n11
»
s21
21
j 4 1 aj )
and the proof of Theorem 3 is complete.
42q
j41
aj
bn 1 1
n11
dn21 1
E1
r
Acknowledgement. I would like to thank Terence Jackson for his
comments on an earlier draft of this paper.
REFERENCES
[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, Acta Arith., LXIII (1993), pp. 313-323.
.
Irrational rapidly convergent series
231
[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. ERDÖS, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci., 10 (1975), pp. 1-7.
[5] J. HANČL J., Criterion for irrational sequences, J. Number Theory, 43, no. 1
(1993), pp. 88-92.
[6] K. MAHLER, Lectures on Transcendental numbers, Lecture notes in mathematics 546, Springer, 1976.
[7] K. NISHIOKA, Mahler Functions and Transcendence, Lecture notes in mathematics 1631, Springer, 1996.
[8] J. SÁNDOR, Some classes of irrational numbers, Studia Univ. Babes Bolyai,
Mathematica, 29 (1984), pp. 3-12.
Manoscritto pervenuto in redazione il 25 ottobre 2001.