# Agnew's theorem

Agnew's theorem, proposed by American mathematician Ralph Palmer Agnew, characterizes reorderings of terms of infinite series that preserve convergence for all series.

## Statement

We call a permutation $p:\mathbb {N} \to \mathbb {N}$ an Agnew permutation if there exists $K\in \mathbb {N}$ such that any interval that starts with 1 is mapped by p to a union of at most K intervals, i.e., ∃K∈N:∀n∈N#$_{}$(p([1, n]))≤K{\textstyle \exists K\in \mathbb {N} \, :\;\forall n\in \mathbb {N} \;\;\#_{[\, ]}(p([1, \, n]))\leq K\, }, where $\#_{[\,]}$ counts the number of intervals.
Agnew's theorem. $p$ is an Agnew permutation $\iff$ for all converging series of real or complex terms ∑$_{i=1}$$^∞$a$_i${\textstyle \sum _{i=1}^{\infty }a_{i}\, }, the series ∑$_{i=1}$$^∞$a$_{p(i)}${\textstyle \sum _{i=1}^{\infty }a_{p(i)}} converges to the same sum.
Corollary 1. $p^{-1}$ (the inverse of $p$) is an Agnew permutation $\implies$ for all diverging series of real or complex terms ∑$_{i=1}$$^∞$a$_i${\textstyle \sum _{i=1}^{\infty }a_{i}\, }, the series ∑$_{i=1}$$^∞$a$_{p(i)}${\textstyle \sum _{i=1}^{\infty }a_{p(i)}} diverges.
Corollary 2. $p$ and $p^{-1}$ are Agnew permutations $\implies$ for all series of real or complex terms ∑$_{i=1}$$^∞$a$_i${\textstyle \sum _{i=1}^{\infty }a_{i}\, }, the convergence type of the series ∑$_{i=1}$$^∞$a$_{p(i)}${\textstyle \sum _{i=1}^{\infty }a_{p(i)}} is the same.

## Usage

Agnew's theorem is useful when the convergence of ∑$_{i=1}$$^∞$a$_i${\textstyle \sum _{i=1}^{\infty }a_{i}} has already been established: any Agnew permutation can be used to rearrange its terms while preserving convergence to the same sum.
The Corollary 2 is useful when the convergence type of ∑$_{i=1}$$^∞$a$_i${\textstyle \sum _{i=1}^{\infty }a_{i}} is unknown: the convergence type of ∑$_{i=1}$$^∞$a$_{p(i)}${\textstyle \sum _{i=1}^{\infty }a_{p(i)}} is the same as that of the original series.

## Examples

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

### Bounded groups of consecutive terms

When the sizes of all groups of consecutive terms are bounded by a constant, i.e., $g_{k+1}-g_{k}\leq L\,$, $p$ and its inverse are Agnew permutations (with K=⌊ L/2 ⌋{\textstyle K=\left\lfloor {\frac {L}{2}}\right\rfloor }), 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 $p_{k}$.
Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval $[1,\,g_{k}]$, hence $p$ and its inverse are Agnew permutations (with $K=2$), i.e., mirroring and circular shifting can be applied within the groups with the convergence type preserved.
A block reordering permutation with B > 1 blocks and its inverse add at most ⌈ B/2 ⌉{\textstyle \left\lceil {\frac {B}{2}}\right\rceil } intervals (when g$_{k+1}$−g$_k${\textstyle g_{k+1}-g_{k}} is large) to the main interval $[1,\,g_{k}]$, hence $p$ and its inverse are Agnew permutations, i.e., block reordering can be applied within the groups with the convergence type preserved.
* A permutation $p_{k}$ mirroring the elements of its interval $[g_{k}+1,\,g_{k+1}]$
* A permutation $p_{k}$ circularly shifting to the right by 2 positions the elements of its interval $[g_{k}+1,\,g_{k+1}]$
* A permutation $p_{k}$ reordering the elements of its interval $[g_{k}+1,\,g_{k+1}]$ as three blocks