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.^[1]

We call a permutation ${\displaystyle p:\mathbb {N} \to \mathbb {N} }$ an Agnew permutation^[a] if there exists ${\displaystyle K\in \mathbb {N} }$ such that any interval that starts with 1 is mapped by p to a union of at most K intervals, i.e., ${\textstyle \exists K\in \mathbb {N} \,:\;\forall n\in \mathbb {N} \;\;\#_{[\,]}(p([1,\,n]))\leq K\,}$, where ${\displaystyle \#_{[\,]}}$ counts the number of intervals.

Agnew's theorem.  ${\displaystyle p}$ is an Agnew permutation ${\displaystyle \iff }$ for all converging series of real or complex terms ${\textstyle \sum _{i=1}^{\infty }a_{i}\,}$, the series ${\textstyle \sum _{i=1}^{\infty }a_{p(i)}}$ converges to the same sum.^[2]

Corollary 1.  ${\displaystyle p^{-1}}$ (the inverse of ${\displaystyle p}$) is an Agnew permutation ${\displaystyle \implies }$ for all diverging series of real or complex terms ${\textstyle \sum _{i=1}^{\infty }a_{i}\,}$, the series ${\textstyle \sum _{i=1}^{\infty }a_{p(i)}}$ diverges.^[b]

Corollary 2.  ${\displaystyle p}$ and ${\displaystyle p^{-1}}$ are Agnew permutations ${\displaystyle \implies }$ for all series of real or complex terms ${\textstyle \sum _{i=1}^{\infty }a_{i}\,}$, the convergence type of the series ${\textstyle \sum _{i=1}^{\infty }a_{p(i)}}$ is the same.^[c][b]

Agnew's theorem is useful when the convergence of ${\textstyle \sum _{i=1}^{\infty }a_{i}}$ 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 ${\textstyle \sum _{i=1}^{\infty }a_{i}}$ is unknown: the convergence type of ${\textstyle \sum _{i=1}^{\infty }a_{p(i)}}$ is the same as that of the original series.

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

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

When the sizes of groups of consecutive terms grow without bounds, it is necessary to look at the behavior of ${\displaystyle p_{k}}$.

Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval ${\displaystyle [1,\,g_{k}]}$, hence ${\displaystyle p}$ and its inverse are Agnew permutations (with ${\displaystyle K=2}$), 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^[d] and its inverse add at most ${\textstyle \left\lceil {\frac {B}{2}}\right\rceil }$ intervals (when ${\textstyle g_{k+1}-g_{k}}$ is large) to the main interval ${\displaystyle [1,\,g_{k}]}$, hence ${\displaystyle p}$ and its inverse are Agnew permutations, i.e., block reordering can be applied within the groups with the convergence type preserved.

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  A permutation ${\displaystyle p_{k}}$ mirroring the elements of its interval ${\displaystyle [g_{k}+1,\,g_{k+1}]}$
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  A permutation ${\displaystyle p_{k}}$ circularly shifting to the right by 2 positions the elements of its interval ${\displaystyle [g_{k}+1,\,g_{k+1}]}$
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  A permutation ${\displaystyle p_{k}}$ reordering the elements of its interval ${\displaystyle [g_{k}+1,\,g_{k+1}]}$ as three blocks