Are there Generalisations of a Limit (for Just-divergent Sequences)? - MathOverflow

Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-12T23:37:45Z

There are certain sequences such as

0, 1, 0, 1, 0, 1, 0, 1, ...

that do not converge, but that may be assigned a generalised limit. Such a sequence is said to diverge, although in this case a phrase such as has an orbit might be preferable.

One way to generalise a limit is by considering the sequence of accumulated means: given a sequence

a₁, a₂, a₃, a₄, ...

the accumulated mean sequence would be

a₁, (a₁+a₂)/2, (a₁+a₂+a₃)/3, (a₁+a₂+a₃+a₄)/4, ...

If this sequence has a limit, then the original sequence may be said to have that value as its generalised limit. In this way, the example sequence above has the generalised limit of 1/2; this seems natural as the sequence oscillates around this 'mean' value.

Is there a name for this kind of generalised limit? Are there other ways to define such a thing. Do you know of any good on-line references for this?

Thanks.

Answer by Andy Putman for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-12T23:51:06Z

A nice book on this kind of stuff is "Classical and modern methods in summability" by Boos and Cass.

Answer by Qiaochu Yuan for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-13T00:29:05Z

Another common technique is Abel summation, which works a little better than Cesaro summation. Zeta regularization is also important in physics.

You might enjoy reading these posts at The Everything Seminar and this column from John Baez.

Answer by Armin Straub for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-13T02:31:45Z

On less practical terms, you can assign a(n extended) limit to any bounded sequence once you have an ultrafilter (on the natural numbers) at hand: Let F be your ultrafilter (that's what makes it less practical). Then for any bounded sequence x_n there exists a unique x such that for all ε>0 the set {n: |x_n-x|<ε} is contained in F. Define this x to be the limit of x_n.

For your sequence 0,1,0,1,... this will assign either 0 or 1 as the limit depending on whether the chosen ultrafilter contains the set of even or the set of odd natural numbers.

This extended notion of limit still is

an algebra homomorphism (from bounded sequences to numbers),
is bounded (ie. takes its value between the infimum and supremum of the sequence), and
is non-principal (that is sequences differing at finitely many indices only get assigned the same limit).

Note that boundedness and non-principality alone suffice to show that for convergent sequences (in the usual sense) we don't get anything new: the extended notion agrees with the classical one.

Of course, there's something to be sacrificed: the extended limit will, for instance, no longer be shift-invariant (meaning that x_n and x_n+h may have different limits).

More details can be found in the following very informal handout I wrote for a student colloquium talk a few years ago. I also very much recommend Terry Tao's related blog post.

Answer by Ari Shnidman for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-13T02:56:34Z

Cesaro summation (the process which you describe) defines a linear functional on a subspace of the Banach space of bounded sequences (namely those sequences which are cesaro summable). Using Hahn-Banach (or one of its variants), one can extend this linear functional to the whole space of bounded sequences, and the extension WILL be shift invariant. However, the extension is not unique and existence depends on the Axiom of choice.

See the Wikipedia entry for Banach limit for more info.

Answer by Jose Brox for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2009-11-13T10:10:07Z

A good site (other than Wikipedia) for summation methods is the Encyclopaedia of Mathematics of SpringerLink. You can start at:

http://eom.springer.de/s/s091140.htm

And then look at Cesàro, Abel, Borel and matrix summations methods for an introduction (but you have many more! There there are Voronoi, Lindëlof, Riesz, Hölder...).

Answer by Jose javier Garcia Moreta for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2010-02-04T20:51:04Z

you can take a look to these papers

http://wbabin.net/science/moreta23.pdf

Author explain in a simple fashion divergent series.

Answer by sheldon-cooper for Are there Generalisations of a Limit (for Just-divergent Sequences)?

2010-02-04T21:23:00Z

Another possibility is to look at how the values are distributed and see whether that converges to some distribution. This is mostly used in stochastic series (e.g. people want to construct Markov chains that converge to a certain distribution of interest).