Agnew's theorem explained
Agnew's theorem characterizes term rearrangements that preserve convergence of series. It was proposed by American mathematician Ralph Palmer Agnew.
Statement
Let be a permutation of
, i.e., a
bijective function
. Then the following two statements are equivalent:
[1] - For any convergent series of real or complex terms , the series converges to the same sum.
- There exists a constant such that, for any
, maps the interval to a union of at most intervals.
Examples
Let us split
in intervals:
where
and
for any
.
Let us also consider a permutation
p=p1\circ … \circpk\circ …
composed of an infinite number of permutations
that permute numbers within corresponding intervals:
Since each
maps
to itself, it follows that
maps
to:
- itself, if
for some
, or
- the union of
and the image under
of
, if
for some
.
Hence, the total number of intervals in the image under
of
equals 1 plus whatever number of additional intervals is created by
.
Bounded intervals
Permutation
can create at most
additional intervals by mapping the first half of its interval,
, in an interleaving fashion:
If the lengths of the intervals are bounded, i.e.,
, then permutation
can create at most
additional intervals, fulfilling the criterion in Agnew's theorem. Therefore, any
may be used.
This means that the terms of any convergent series may be rearranged freely within groups, if the lengths of these groups are bounded by a constant.
Unbounded intervals
Permutations
that mirror their interval:
permutations
that perform right
circular shifts of their interval by
positions (
):
and permutations
that are the inverses of the interleaving permutations described above:
all create 1 additional interval, fulfilling the criterion in Agnew's theorem.
Permutations
that rearrange their interval as
blocks can create at most
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 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
as described above with intervals increasing and
being interleaving permutations described above. Such
does not fulfill the criterion in Agnew's theorem, therefore, there exists a convergent series
such that
is either divergent or converges to a different sum. But it can't converge to a different sum: the inverse permutation
is composed of inverses of interleaving permutations
, which all fulfill the criterion in Agnew's theorem, therefore
would converge to the same sum as
. This means that
must be divergent.
However, if we require both
and
to satisfy the criterion in Agnew's theorem, then
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
, such that both
and
map
to at most
intervals, without changing the type of convergence/divergence of the series.
Notes and References
- Ralph Palmer . Agnew . 1955 . Permutations preserving convergence of series . Proc. Amer. Math. Soc. . 6 . 4 . 563–564.