View source for Agnew's theorem

{{short description|Theorem about permutations that preserve convergence for all converging series}}

'''Agnew's theorem''', proposed by American mathematician [[Ralph Palmer Agnew]], characterizes reorderings of terms of infinite [[series (mathematics)|series]] that preserve [[convergent series|convergence]] for all series.<ref>{{cite journal |first=Paul |last=Schaefer |date=1981 |title=Sum-preserving rearrangements of infinite series |url=https://www.math.hkust.edu.hk/~mamyan/research/UROP/schaefer.pdf |journal=Amer. Math. Monthly |volume=88 |issue=1 |pages=33-40}}</ref>

== Statement ==

We call a [[permutation]] <math>p: \mathbb{N} \to \mathbb{N}</math> an ''Agnew permutation''{{efn|This terminology is used only in this article, to simplify the explanation.}} if there exists <math>K \in \mathbb{N}</math> such that any interval that starts with 1 is mapped by {{math|''p''}} to a union of at most {{math|''K''}} intervals, i.e., <math display=inline>\exists K \in \mathbb{N} \, : \; \forall n \in \mathbb{N} \;\; \#_{[\,]}(p([1,\,n])) \le K\,</math>, where <math>\#_{[\,]}</math> counts the number of intervals.

'''Agnew's theorem.''' &nbsp;<math>p</math> is an Agnew permutation <math>\iff</math> for all converging series of real or complex terms <math display=inline>\sum_{i=1}^\infty a_i\,</math>, the series <math display=inline>\sum_{i=1}^\infty a_{p(i)}</math> converges to the same sum.<ref>{{cite journal |first=Ralph Palmer |last=Agnew |date=1955 |title=Permutations preserving convergence of series |url=https://www.ams.org/journals/proc/1955-006-04/S0002-9939-1955-0071559-4/S0002-9939-1955-0071559-4.pdf |journal=Proc. Amer. Math. Soc. |volume=6 |issue=4 |pages=563–564}}</ref>

'''Corollary 1.''' &nbsp;<math>p^{-1}</math> (the inverse of <math>p</math>) is an Agnew permutation <math>\implies</math> for all diverging series of real or complex terms <math display=inline>\sum_{i=1}^\infty a_i\,</math>, the series <math display=inline>\sum_{i=1}^\infty a_{p(i)}</math> diverges.{{efn|name=imp|Note that, unlike Agnew's theorem, the corollaries in this article do not specify equivalence, only implication.}}

'''Corollary 2.''' &nbsp;<math>p</math> and <math>p^{-1}</math> are Agnew permutations <math>\implies</math> for all series of real or complex terms <math display=inline>\sum_{i=1}^\infty a_i\,</math>, the convergence type of the series <math display=inline>\sum_{i=1}^\infty a_{p(i)}</math> is the same.{{efn|Absolutely converging series turn into absolutely converging series, conditionally converging series turn into conditionally converging series (with the same sum), diverging series turn into diverging series.}}{{efn|name=imp}}

== Usage ==

Agnew's theorem is useful when the convergence of <math display=inline>\sum_{i=1}^\infty a_i</math> has already been established: any Agnew permutation can be used to rearrange its terms while preserving convergence to the same sum.

The Corollary 2 is useful when the convergence type of <math display=inline>\sum_{i=1}^\infty a_i</math> is unknown: the convergence type of <math display=inline>\sum_{i=1}^\infty a_{p(i)}</math> is the same as that of the original series.

== Examples ==

An important class of permutations is infinite compositions of permutations <math>p=\cdots \circ p_k \circ \cdots \circ p_1</math> in which each constituent permutation <math>p_k</math> acts only on its corresponding interval <math>[g_k+1,\,g_{k+1}]</math> (with <math>g_1=0</math>). Since <math>p([1,\,n]) = [1,\,g_k] \cup p_k([g_k+1,\,n])</math> for <math>g_k+1 \le n < g_{k+1}</math>, we only need to consider the behavior of <math>p_k</math> as <math>n</math> increases.

=== Bounded groups of consecutive terms ===

When the sizes of all groups of consecutive terms are bounded by a constant, i.e., <math>g_{k+1}-g_k \le L\,</math>, <math>p</math> and its inverse are Agnew permutations (with <math display=inline>K = \left\lfloor\frac{L}{2}\right\rfloor</math>), i.e., arbitrary reorderings can be applied within the groups with the convergence type preserved.

=== Unbounded groups of consecutive terms ===

When the sizes of groups of consecutive terms grow without bounds, it is necessary to look at the behavior of <math>p_k</math>.

Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval <math>[1,\,g_k]</math>, hence <math>p</math> and its inverse are Agnew permutations (with <math>K = 2</math>), i.e., mirroring and circular shifting can be applied within the groups with the convergence type preserved.

A block reordering permutation with {{math|''B''}}&nbsp;>&nbsp;1 blocks{{efn|1=The case of {{math|''B''}}&nbsp;=&nbsp;2 is a circular shift.}} and its inverse add at most <math display=inline>\left\lceil\frac{B}{2}\right\rceil</math> intervals (when <math display=inline>g_{k+1}-g_k</math> is large) to the main interval <math>[1,\,g_k]</math>, hence <math>p</math> and its inverse are Agnew permutations, i.e., block reordering can be applied within the groups with the convergence type preserved.

<gallery mode=packed style="text-align:left">
A permutation mirroring the elements of an interval.svg|A permutation <math>p_k</math> mirroring the elements of its interval <math>[g_k+1,\,g_{k+1}]</math>
A permutation circularly shifting the elements of an interval.svg|A permutation <math>p_k</math> circularly shifting to the right by 2 positions the elements of its interval <math>[g_k+1,\,g_{k+1}]</math>
A permutation reordering the elements of an interval as 3 blocks.svg|A permutation <math>p_k</math> reordering the elements of its interval <math>[g_k+1,\,g_{k+1}]</math> as three blocks
</gallery>

== Notes ==

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== References ==

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[[Category:Mathematical theorems]]