Abstract
AbstractAssume that a convergent series of real numbers $$\sum \limits _{n=1}^\infty a_n$$
∑
n
=
1
∞
a
n
has the property that there exists a set $$A\subseteq {\mathbb {N}}$$
A
⊆
N
such that the series $$\sum \limits _{n \in A} a_n$$
∑
n
∈
A
a
n
is conditionally convergent. We prove that for a given arbitrary sequence $$(b_n)$$
(
b
n
)
of real numbers there exists a permutation $$\sigma :{\mathbb {N}}\rightarrow {\mathbb {N}}$$
σ
:
N
→
N
such that $$\sigma (n) = n$$
σ
(
n
)
=
n
for every $$n \notin A$$
n
∉
A
and $$(b_n)$$
(
b
n
)
is $$c_0$$
c
0
-equivalent to a subsequence of the sequence of partial sums of the series $$\sum \limits _{n=1}^\infty a_{\sigma (n)}$$
∑
n
=
1
∞
a
σ
(
n
)
. Moreover, we discuss a connection between our main result with the classical Riemann series theorem.
Publisher
Springer Science and Business Media LLC
Subject
Control and Optimization,Analysis,Algebra and Number Theory
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