Agnew's theorem

Agnew's theorem characterizes term rearrangements that preserve convergence of series. It was proposed by American mathematician Ralph Palmer Agnew.

Statement

Let p be a permutation of $\mathbb {N}$, i.e., a bijective function $p:\mathbb {N} \to \mathbb {N}$. Then the following two statements are equivalent:

1. For any convergent series of real or complex terms ${\textstyle \sum _{n=1}^{\infty }a_{n}}$, the series ${\textstyle \sum _{n=1}^{\infty }a_{p(n)}}$ converges to the same sum.
2. There exists a constant K such that, for any $n\in \mathbb {N}$, p maps the interval [1, n] to a union of at most K intervals.

Examples

Let us split $\mathbb {N}$ in intervals:

$$[g_{0}+1,\,g_{1}],\,\ldots ,\,[g_{k-1}+1,\,g_{k}],\,\ldots \;,$$

where $g_{0}=0$ and $g_{k}>g_{k-1}$ for any $k\in \mathbb {N}$.

Let us also consider a permutation $p=p_{1}\circ \cdots \circ p_{k}\circ \cdots$ composed of an infinite number of permutations $p_{k}$ that permute numbers within corresponding intervals:

$${\begin{cases}p_{k}(n)\in [g_{k-1}+1,\,g_{k}]&{\text{if}}\;\;n\in [g_{k-1}+1,\,g_{k}]\\p_{k}(n)=n&{\text{otherwise}}\end{cases}}$$

Since each $p_{k}$ maps $[g_{k-1}+1,\,g_{k}]$ to itself, it follows that $p$ maps $[1,n]$ to:

1. itself, if $n=g_{k}$ for some $k$, or
2. the union of $[1,\,g_{k-1}]$ and the image under $p_{k}$ of $[g_{k-1}+1,\,n]$, if $n\in [g_{k-1}+1,\,g_{k}-1]$ for some $k$.

Hence, the total number of intervals in the image under $p$ of $[1,n]$ equals 1 plus whatever number of additional intervals is created by $p_{k}$.

Bounded intervals

Permutation $p_{k}$ can create at most ${\textstyle \left\lfloor {\frac {g_{k}-g_{k-1}}{2}}\right\rfloor }$ additional intervals by mapping the first half of its interval, ${\textstyle [g_{k-1}+1,\,g_{k-1}+\left\lfloor {\frac {g_{k}-g_{k-1}}{2}}\right\rfloor ]}$, in an interleaving fashion:

$$p_{k}(g_{k-1}+n)=g_{k-1}+2n\;.$$

If the lengths of the intervals are bounded, i.e., $g_{k}-g_{k-1}\leq L$, then permutation $p_{k}$ can create at most ${\textstyle \left\lfloor {\frac {L}{2}}\right\rfloor }$ additional intervals, fulfilling the criterion in Agnew's theorem. Therefore, any $p_{k}$ may be used.

This means that the terms of any convergent series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ may be rearranged freely within groups, if the lengths of these groups are bounded by a constant.

Unbounded intervals

Permutations $p_{k}$ that mirror their interval:

$$p_{k}(g_{k-1}+n)=g_{k}+1-n\;,$$

permutations $p_{k}$ that perform right circular shifts of their interval by $S$ positions ($0<S<g_{k}-g_{k-1}$):

$$p_{k}(g_{k-1}+n)=g_{k-1}+1+\left((n-1+S){\bmod {(}}g_{k}-g_{k-1})\right)\;,$$

and permutations $p_{k}$ that are the inverses of the interleaving permutations described above:

$$p_{k}(g_{k-1}+n)={\begin{cases}g_{k-1}+\left\lfloor {\frac {g_{k}-g_{k-1}}{2}}\right\rfloor +{\frac {n+1}{2}}&{\text{if}}\;n\;{\text{odd}}\\g_{k-1}+{\frac {n}{2}}&{\text{if}}\;n\;{\text{even}}\end{cases}}$$

all create 1 additional interval, fulfilling the criterion in Agnew's theorem.

Permutations $p_{k}$ that rearrange their interval as $B>1$ blocks can create at most ${\textstyle \min(\left\lceil {\frac {B}{2}}\right\rceil ,\left\lfloor {\frac {g_{k}-g_{k-1}}{2}}\right\rfloor )}$ additional intervals. If the number of these blocks is bounded, then the criterion in Agnew's theorem is fulfilled.

This means that within groups of arbitrary unbounded length the terms of any convergent series ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ may be mirrored, circularly shifted and rearranged in blocks (if the number of these blocks is bounded by a constant); terms at even positions within groups may be gathered at the beginning of the group (in the same order).

Dealing with unknown series

The permutations described by Agnew's theorem can transform a divergent series into a convergent one. Let us consider a permutation $p$ as described above with intervals increasing and $p_{k}$ being interleaving permutations described above. Such $p$ does not fulfill the criterion in Agnew's theorem, therefore, there exists a convergent series ${\textstyle \sum _{i=1}^{\infty }a_{n}}$ such that ${\textstyle \sum _{i=1}^{\infty }a_{p(n)}}$ is either divergent or converges to a different sum. But it can't converge to a different sum: the inverse permutation $p^{-1}$ is composed of inverses of interleaving permutations $p_{k}^{-1}$, which all fulfill the criterion in Agnew's theorem, therefore ${\textstyle \sum _{i=1}^{\infty }a_{p^{-1}(p(n))}=\sum _{i=1}^{\infty }a_{n}}$ would converge to the same sum as ${\textstyle \sum _{i=1}^{\infty }a_{p(n)}}$. This means that ${\textstyle \sum _{i=1}^{\infty }a_{p(n)}}$ must be divergent.

However, if we require both $p$ and $p^{-1}$ to satisfy the criterion in Agnew's theorem, then $p$ will preserve both convergence (with the same sum) and divergence. (If it didn't preserve divergence, then the inverse wouldn't preserve convergence.)

In fact, such permutations preserve absolute convergence (with the same sum), conditional convergence (with the same sum) and divergence. (All permutations preserve absolute convergence with the same sum; a conditionally convergent series can't be turned into an absolutely convergent one because the reverse permutation wouldn't preserve absolute convergence.)

This means that, when dealing with a series for which it is unknown whether it converges and what type of convergence it has, its terms may be rearranged using permutations $p$, such that both $p$ and $p^{-1}$ map $[1,n]$ to at most $K$ intervals, without changing the type of convergence/divergence of the series.