Author: Isaac Zylstra
Calvin College
Ratio Series
Ratio series are series for which the ration an+1 /
an is bounded, either above or below by a
constant have interesting properties.
1 an+1
If ≤
< 1 for every n, then the set of
2 an
selective sums is [0,s].
an+1 1
If 0 <
≤ , then the set of selective sums is
an 2
totally disconnected.
Sets of Selective Sums of Infinite Series
What is a selective sum?
∞
𝑛=1 𝑎𝑛 be a series. A number x is a selective sum of the series
= ∞
𝑛=1 𝑐𝑛𝑎𝑛, where {cn} is a sequence consisting only of 0's
Let
if x
and 1's. That is, a selective sum is a sum of the terms of a
subsequence of the sequence {an}.
The set of selective sums of an infinite series is the set of all
possible selective sums for the series ∞
𝑛=1 𝑎𝑛.
Representations
Different sets of selective sums have varying
numbers of representations for each of its
selective sums.
If an > Rn for all n, then every selective sum has
a unique representation.
If an < Rn for infinitely n, and there are
infinitely many n such that the selective sum x
𝑛
𝑛
is in both [ 𝑖=1 𝑐𝑖𝑎𝑖 , 𝑖=1 𝑐𝑖𝑎𝑖 + Rn+1] and
[ 𝑛𝑖=1 𝑐𝑖𝑎𝑖 + an + 1, 𝑛𝑖=1 𝑐𝑖𝑎𝑖 + Rn], then there
are infinitely many representations of x as a
selective sum.
Construction of the set
One way of constructing the set is to do a Cantor-like construction.
This construction is called "Cantor-like" because the construction is
similar to the way the Cantor Set is constructed.
∞
𝑛=1 𝑎𝑛
Alternating Series
as a convergent series of strictly
positive terms and strictly decreasing terms.
𝑛+1
Then ∞
(−1)
𝑎𝑛 is our alternating series.
𝑛=1
∞
∞
Suppose that 𝑛=1 𝑎2𝑛−1 = s1 and 𝑛=1 𝑎2𝑛 =
s 2.
an+1
1
If
≥ for all n, then the set of selective of
an
√2
the alternating series is [-s2, s1]
an+1 1
If
< , then S is equal to the difference of
an √2
two totally disconnected sets.
Let ∞
𝑛=1 𝑎𝑛 be a convergent series with sum s, and with terms that
are strictly positive and strictly decreasing. Let Rn = ∞
𝑖=𝑛+1 𝑎𝑖 .
Begin with the closed interval [0,s], and let F1 = [0, R1] ∪ [s-R1, s]
= [0, R1 ∪ [a1, s]. Then let Fn+1 be the union of all intervals [x,
x+Rn] and [y-Rn, y], where [x, y] is an interval in Fn. Then
∞
∞
𝐹
is
equal
to
the
set
of
selective
sums
of
the
series
𝑛=1 𝑛
𝑛=1 𝑎𝑛.
Five Rules
This construction leads to a set of rules which are the basis for our
results. These rules are:
1. If an ≤ Rn for all n, then the set of selective sums is [0, ∞
𝑛=1 𝑎𝑛]
2. If an < Rn for infinitely many n, but an > Rn for finitely many n,
then the set of selective sums is a union of intervals on [0,
∞
𝑛=1 𝑎𝑛].
3. If an > Rn for infinitely many n, and an < Rn for infinitely many n,
it is possible to get a union of totally disconnected sets, but
beyond that, we still don't know (more on this below).
4. If an > Rn for infinitely many n, but an < Rn for finitely many n,
then the set of selective sums is an union of totally disconnected
sets on [0, ∞
𝑛=1 𝑎𝑛].
5. If an > Rn for all n, then the set of selective sums is totally
disconnected on [0, ∞
𝑛=1 𝑎𝑛].
Open Questions
We were unable to solve all of the problems we
can across. 3 problems in particular seemed
interesting:
1. Rule 3, as mentioned earlier: Can an interval
could be contained with the set of selective
sums of a series with an > Rn for infinitely many
n, and an < Rn for infinitely many n?
2. Uncountably infinite representations: Is it
possible to construct a series whose set of
selective sums contains a selective sum with an
uncountable infinite number of representations?
3. 2-dimensional selective sums: What are the
properties of selective sums of series of the
∞
form 𝑛=1 𝑎𝑛𝑥 + 𝑏𝑛𝑦?