Pacific Journal of
Mathematics
REPRESENTATION OF REAL NUMBERS BY GENERALIZED
GEOMETRIC SERIES
E UGENE A. M AIER
Vol. 28, No. 3
May 1969
PACIFIC JOURNAL OF MATHEMATICS
Vol. 28, No. 3, 1969
REPRESENTATION OF REAL NUMBERS BY
GENERALIZED GEOMETRIC SERIES
E. A. MAIER
We shall say that the series of real numbers, Σ^=o V^u
is a generalized geometric series (g.g.s.) if and only if
a\ ^ cLi+idi-! for all i ^ 1. (Note that the series is geometric
if and only if equality holds.) In this paper we investigate
the representation of positive real numbers less than or equal
to one by generalized geometric series of the form Σ£=o a^/c*
where the d are positive integers and x ^ 1.
1*
Preliminary results*
LEMMA 1. If Σf=o 1/α* is a g.g.s. and
< \ak+ί\ ,
Proof.
then
1
~ \ak\
i=k+l
Since \ak/ak+t\ iS \ak/ak+1\' for all t ^ 1, we have
Σ
1
^
1 v
\ak+1\ — \ak
\ak
The following theorem readily follows from Lemma 1.
THEOREM 1. The g.g.s. ΣΓ=o 1/α* converges if and only if there
exists k such that \ak\ < | α Λ + 1 | .
THEOREM 2. Let ΣΓ=o l/α f 6e α g.g.s. wΐ£/2, 0 < α 0 < α x . L e i a =
ΣΓ=i I K , S fc = Σf=o l/α* α ^ d ίfc+1 = ak+1/ak - 1. Γ/^β^
(i) the sequence of half-open intervals {(Sk, Sk + l/(α Λ + 1 — α^)]} is
α sequence of nested intervals whose intersection is a,
(ii) tk ^ tk+1 ^ l/ak(a - Sk).
Proof.
Since the series is a g.g.s., we have
1
^
1
+
1
Hence the sequence of intervals in (i) above is nested.
for all k Ξ> 0. Thus, using Lemma 1,
(1)
Sk + ^ — .
Sk < a ^ Sk +
603
Also ak < ak+1
604
E. A. MAIER
Since ak/ak+1 ^ a^ax for all k ^ 0, we have
0 ^ lim
*
-
^ lim
<*
k
&
(CLi
A
(CL
A
k\
~~ -L I
\a0
J
= 0,
π
a
and it follows from (1) that the intervals converge to a.
Inequalities (ii) are obtained from (1) and the definition of tk.
Let x > 0 and let a = ΣΓ=o %%/Ci be a g.g.s.
COROLLARY.
C
α
with
s
0 < cox < d; Zeί S* = Σ*=o #7 * ^ k+ι = Ck+ι/Ck — 1. ΪT^βw
(i) ίfeβ sequence of half-open intervals {(Sk, Sk + xk+1/(ck+ι — xck)]}
is a sequence of nested intervals
whose intersection
is a,
(ii) sk £ sk+1 £ xk+ι/(ck(a - Sk)) + (x- 1).
Apply the theorem with ak = ck/xk observing that in this
Proof.
case
J.
Vjc + 1 —
£fc + l
1±
—
XCk
^A + l ~i~ J-
11
--
X
3. T h e representation of reals/ The corollary to Theorem 2
suggests an algorithm for constructing a g.g.s. of the form X,Γ=o ^{/cu
where the c< are integers and x ^ 1, which converges to a given positive real number β ^ 1.
To obtain such a series let {s0, su s2
•} be any sequence of positive integers such that s0 = [1//5] and, for k ^> 0,
rγk + l
{ —I
c {β
k
}
— - 1, 8* - if < sk+ι £ —I
J
- sk)
Λ-fc + 1
chφ -
— + (x - 1)
sk)
where c, = UU (βy + 1) and S 4 = Σi=o xV^
Such a sequence of integers exists since
ck(β - S t ) = ck^(sk
and hence
+ (a; - 1) ^
c
,f+\,
+(x-l).
The resulting series, ΣΠ=o*V ί where c4 = Π* =o (s, +1), is a g.g.s.
since s ^ ^ s 4 . Also since β ^ 1,
GENERALIZED GEOMETRIC SERIES
605
([1] >
*!
Thus the series satisfies the hypotheses of the corollary to Theorem 2.
Now from the manner in which the sequence {sk} has been obtained,
we have
x
-1 ^
_
<---
ck(β - &)
a?
ι /
Λ
Λ
i \
- cfc(/3 - Sk)
and thus
+ 1)
c Λ (s Λ + 1 + 1 - a?)
zy.fe + 1
= sk + —
.
Therefore, by (i) of the corollary, β = ΣΠ=o a^M
If x Φ 1, the sequence {sΛ} obtained by the above process is not
unique. For example, if β = 1 and x = 2, we have s0 = 1,
maxj
x
lcoβ — 1
- 1, s0 -
= max {1, 0} = 1
and
—-— + x - 1= 3.
Co — 1
Thus there are two possible values for sλ. To obtain uniqueness, we
must further restrict the sk. One restriction that leads to a unique
representation is to require that sk ^> xsk_l9 We now turn our attention to series which satisfy this condition.
3. Let s > 0, x ^ 1. For ft ^ 0, let sk = α;fes, cfc =
(Si + 1). 77*<m Σ"-o xyc* = 1/s; that is
THEOREM
Πί = 0
8
s + 1
(s + l)(a;s + 1)
(s + l)(αs + l)(a;2s + 1)
Proo/. Let S& = Σ ί U a ? ^ . We shall show by induction that
Sk + IMS = 1/s for all ft ^ 0. For ft = 0, we have
If Sfc + l/cAs = 1/s, then
606
E. A. MAIER
1
= sk
= λ - J_ + 11 /
k+1
+1
sx
+ 1) /
s
It also follows by induction that ck > (s + 1)* and hence, since s + 1 > 1,
o,, l/cΛs = 0. Thus lim^*, Sk = l i m ^ (1/s + l/c*s) = 1/s.
tive
T H E O R E M 4.
Let
x ^ l .
integers
that
sk ^ xsk_x
α
convergent
such
ΣΓ-o »VC< ^ s
Let
{s 0 , slf si9
•} be a sequence
of
posi-
and let ck = Π?=o (s< + 1 ) .
for all k^O
g.g.s. Furthermore
if a — ΣΓ=o »Vcί
( i ) the sequence of half-open intervals {(Sk, Sk + xk+ί/(ck+1 — ck)]}
is a sequence of nested intervals whose intersection is a,
k+
(ϋ) Sk+1 = [x '/ck(a - Sk)] for all k ^ 0, s0 = [1/α],
(iii) i / ίc is rational, then a is rational if and only if sk = xsj,^
for all k sufficiently large.
Proof.
Since cjc^
= s< + 1 ^ s i + 1 + 1 = ci+1/cif it follows that
and hence ΣΓ=o # 7 ^ is a g.g.s. The series converges by Theorem 1
since c0 < cjx.
To establish (i), we first observe that sk+j+ί ^ x3'sk+1 for all j ^ 0.
Thus, using Theorem 3, we have
Sk < a = Sk + Σ
t=0
"
+
7
Tΰ——Tw
TV
+ ^——^^-+
( s Λ + 1 + l)(a?Sfc+i + l)(x2sk+1
- sk + ^ i
+
l)(a?sΛ+1
+ 1)
- i - - sk + ^ + 1
Furthermore, since sk+2 — xsk+1 ^ 0, we have
GENERALIZED GEOMETRIC SERIES
"1
607
'V
\
2 '
>k + l /
11
Ίi +
χ
S
C
k +1 ^
-
fc + 1
x /v.
\
) ^
/y.fc + 1
Sfc+2 '
Thus
y.fc + 1
and the sequence of intervals in (i) is nested.
Since
by part (i) of the corollary to Theorem 2, the intersection of the intervals is a.
To establish (ii), we have from (i) that
e*(Sjfc+i + 1)
CkSk+ι
Solving these inequalities for sk+u
( 2 )
ck(a -
we have
X
)
St
Also, since s t ^ ί»*s0, using Theorem 3, we have
a ^
s0 + 1
s0 + 1
(s0 + l)(xsQ + 1)
/v.2
+
1
?
+ ... = JL
and hence so]=Jl/a].
We turn now to the proof of (iii). Suppose sk = xsk^ for all
k > k0. Then sko+j = ^ J s Ao for all j ^ 0. Thus, again using Theorem
3, we have
_ V
o 1
x
(skQ+1)(x8ko
+
+ 1)
2
(sko+1)(%sko+1)(x skQ+1)
608
E. A. MAIER
which is rational if x is rational.
Conversely, suppose a is rational.
From (2) we have
sk+1
Thus
γk + l \
( a(3)
= ek(8k+ί + l ) ( α -Sk)
Sk--—
k+1
-
x
(a — bk)
Hence, if α = p/g, for all k we have
0 < ck+1(p - Sk+1q) ^ ck(p - Skq) .
Therefore, noting that ckSk is an integer, {ck(p — Skq)} is a nonincreasing sequence of positive integers and thus for k sufficiently large, say
k > k0, the terms of the sequence become constant. Hence, for k > k0,
ck+1(a — Sk+1) = ck(a — Sk) and thus equality must hold in (3). Therefore sk+1 = xk+1/ck(a — SΛ) for k > kQ and, for & sufficiently large,
X
=
=
XSk
THEOREM 5. Let 0 < β ^ 1 αmZ ϊeί a? δβ α positive integer. Then
there exists a unique
sequence
of positive
integers
{s 0 , s19 s2 •••} such
that sk ^ xsk^ for k^l
and β = ΣΓ=o # 7 ^ wΛβrβ ^ = Πί : =o (s{ + 1 ) .
Proof.
Define
So —
I -zr , .
Λ
„
.
i
w -
1
,
Ci
1
Co
and, for k > 0,
77^
P7T~ I >
Then, in the same manner as inequality (3) was obtained, we have
ck+ί(β
Thus
- Sk+1) ^ ck(β - Sk) .
GENERALIZED GEOMETRIC SERIES
/y.fc + 1
sk+1 >
609
zy.fc + 1
*
ek(β - Sk)
= χ(
1>
*
~ ck_t(β - Sk-d
ϊl_
)
- l ^ X S
1
-1 .
k
Since s ^ and ^s^. — 1 are integers, it follows that sk+1 ^ xsk.
from the definition of sk+1,
Also
1 < sk+ι
sk+ι <
~ ck(β - Sk)
ck(β - Sk)
Therefore
/y.t + 1
Sk < Sk+1 = Sk
^
Thus from Theorem 4 (i), /S = Σr=o α V^.
quence {sk} follows from Theorem 4 (ii).
Received March 20, 1968.
UNIVERSITY OF OREGON AND
PACIFIC LUTHERAN UNIVERSITY
γk
*
- Sk)
The uniqueness of the se-
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
H. ROYDEN
Stanford University
Stanford, California
J. DUGUNDJI
Department of Mathematics
University of Southern California
Los Angeles, California 90007
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University of Washington
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Pacific Journal of Mathematics
Vol. 28, No. 3
May, 1969
Jon F. Carlson, Automorphisms of groups of similitudes over F3 . . . . . . . . . . . .
W. Wistar (William) Comfort, Neil Hindman and Stelios A. Negrepontis,
F 0 -spaces and their product with P-spaces. . . . . . . . . . . . . . . . . . . . . . . . . . .
Archie Gail Gibson, Triples of operator-valued functions related to the unit
circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David Saul Gillman, Free curves in E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E. A. Heard and James Howard Wells, An interpolation problem for
subalgebras of H ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Albert Emerson Hurd, A uniqueness theorem for weak solutions of symmetric
quasilinear hyperbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E. W. Johnson and J. P. Lediaev, Representable distributive Noether
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David G. Kendall, Incidence matrices, interval graphs and seriation in
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Robert Leroy Kruse, On the join of subnormal elements in a lattice . . . . . . . . .
D. B. Lahiri, Some restricted partition functions; Congruences modulo 3 . . . .
Norman D. Lane and Kamla Devi Singh, Strong cyclic, parabolic and conical
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Daniel Paul Maki, A note on recursively defined orthogonal polynomials . . . .
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James R. McLaughlin and Justin Jesse Price, Comparison of Haar series with
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William H. Mills and Neal Zierler, On a conjecture of Golomb . . . . . . . . . . . . .
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