Agnew's theorem

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

Agnew's theorem.  $p$ is an Agnew permutation $\backslash iff$ for all converging series of real or complex terms $\backslash sum\_\{i=1\}^\backslash infty\; a\_i\backslash ,$, the series $\backslash sum\_\{i=1\}^\backslash infty\; a\_\{p(i)\}$ converges to the same sum.{{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}}

Corollary 1.  $p^\{-1\}$ (the inverse of $p$) is an Agnew permutation $\backslash implies$ for all diverging series of real or complex terms $\backslash sum\_\{i=1\}^\backslash infty\; a\_i\backslash ,$, the series $\backslash sum\_\{i=1\}^\backslash infty\; a\_\{p(i)\}$ diverges.{{efn|name=imp|Note that, unlike Agnew's theorem, the corollaries in this article do not specify equivalence, only implication.}}

Corollary 2.  $p$ and $p^\{-1\}$ are Agnew permutations $\backslash implies$ for all series of real or complex terms $\backslash sum\_\{i=1\}^\backslash infty\; a\_i\backslash ,$, the convergence type of the series $\backslash sum\_\{i=1\}^\backslash infty\; a\_\{p(i)\}$ 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}}

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

An important class of permutations is infinite compositions of permutations $p=\backslash cdots\; \backslash circ\; p\_k\; \backslash circ\; \backslash cdots\; \backslash circ\; p\_1$ in which each constituent permutation $p\_k$ acts only on its corresponding interval $[g\_k+1,\backslash ,g\_\{k+1\}]$ (with $g\_1=0$). Since $p([1,\backslash ,n])\; =\; [1,\backslash ,g\_k]\; \backslash cup\; p\_k([g\_k+1,\backslash ,n])$ for , we only need to consider the behavior of $p\_k$ as $n$ increases.

When the sizes of all groups of consecutive terms are bounded by a constant, i.e., $g\_\{k+1\}-g\_k\; \backslash le\; L\backslash ,$, $p$ and its inverse are Agnew permutations (with $K\; =\; \backslash left\backslash lfloor\backslash frac\{L\}\{2\}\backslash right\backslash rfloor$), 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 $p\_k$.

Mirroring permutations and circular shift permutations, as well as their inverses, add at most 1 interval to the main interval $[1,\backslash ,g\_k]$, 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 {{math|B}} > 1 blocks{{efn|1=The case of {{math|B}} = 2 is a circular shift.}} and its inverse add at most $\backslash left\backslash lceil\backslash frac\{B\}\{2\}\backslash right\backslash rceil$ intervals (when $g\_\{k+1\}-g\_k$ is large) to the main interval $[1,\backslash ,g\_k]$, 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 mirroring the elements of an interval.svg|A permutation $p\_k$ mirroring the elements of its interval $[g\_k+1,\backslash ,g\_\{k+1\}]$

A permutation circularly shifting the elements of an interval.svg|A permutation $p\_k$ circularly shifting to the right by 2 positions the elements of its interval $[g\_k+1,\backslash ,g\_\{k+1\}]$

A permutation reordering the elements of an interval as 3 blocks.svg|A permutation $p\_k$ reordering the elements of its interval $[g\_k+1,\backslash ,g\_\{k+1\}]$ as three blocks

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Category:Theorems in mathematical analysis

Category:Series (mathematics)