Riemann’s theorem on rearrangements

If the map nn is a bijection on , we say that the sequence (an) is a rearrangement of (an).

The following theorem, which is due to Riemann, shows that the convergence of a conditionally convergent series depends so much on the order of its terms; in particular, a conditionally convergent series can be made to converge to any real number by changing the order of its terms.

Theorem (Riemann series theoremMathworldPlanetmath). Let (an) be a sequence in such that n=1an converges but n=1|an|=, i.e, an is conditionally convergent. Let -α<β be arbitrary. Then there exists a rearrangement (an) such that

lim infNn=1Nan=α   and   lim supNn=1Nan=β.

Proof. Let an+=max{0,an} and an-=min{0,-an}. Then we have an=an+-an- and |an|=an++an-. Since an<, both an+ and an- diverge or converge simultaneously. But since |an|=, we see that at least one of an+ and an- must diverge. It follows that an+= and an-=.

Also by the nth term test, limnan+=limnan-=0.

Now we pass to subsequence of (an+) by removing all terms with an+=0 and an0. For an-, we remove all terms with an-=0. Let us denote the subsequences still as (an+) and (an-). Since only zeros have been removed, an+ and an- are still divergent.

Now we will define integers mj and kj for j, and consider the series

a1+ +a2+++am1+

This series is clearly a rearrangement of an, by our choice of the subsequences an+ and an-.

We pick up two sequences αj and βj such that αjα , βjβ, αnβn and β1>0. We choose m1 such that n=1m1β1 but n=1m1-1<β1. We choose k1 such that n=1m1-n=1k1α1 but n=1m1-n=1k1-1>α1. We continue this way, inductively.

Since limnan+=limnan-=0, the subsequences of the sequence of partial sums that end with amj+ and akj- converge to β and α, and it can be seen that no subsequence can be found with a limit larger than β or lower than α.


Title Riemann’s theorem on rearrangements
Canonical name RiemannsTheoremOnRearrangements
Date of creation 2013-03-22 17:31:52
Last modified on 2013-03-22 17:31:52
Owner Gorkem (3644)
Last modified by Gorkem (3644)
Numerical id 23
Author Gorkem (3644)
Entry type Theorem
Classification msc 40A05
Synonym Riemann series theorem
Related topic UnconditionallyConvergent
Related topic FiniteChangesInConvergentSeries
Related topic FiniteChangesInConvergentSeries2