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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?


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Your example is the sequence of partial sums of the series (-1)^n, which is Cesaro summable to 1/2: see – Steven Sivek Nov 12 '09 at 23:47
There are much worse things you can read than G.H. Hardy's `Divergent Series'. – Mariano Suárez-Alvarez Nov 12 '09 at 23:51
You can read Hardy's book online at – lhf Nov 13 '09 at 0:24
Hardy's book is very good. I would have listed it as an answer, but really Mariano/lhf should, and I can vote their answers up. – Theo Johnson-Freyd Nov 13 '09 at 3:24
From the preface of Hardy's book: "Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever". Attributed to Abel. – Kevin Buzzard Nov 13 '09 at 9:57
up vote 6 down vote accepted

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.

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Another approach (no pun intended) to the limit of sums is Borel Summation – Dan Piponi Nov 13 '09 at 1:19

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 xn there exists a unique x such that for all ε>0 the set {n: |xn-x|<ε} is contained in F. Define this x to be the limit of xn.

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 xn and xn+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.

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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.

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Nice alternative. Of course, while gaining shift-invariance, you have to sacrifice the algebra homomorphism property (the limit is still linear though). On the other hand, that's already true for Cesaro summation... – Armin Straub Nov 13 '09 at 3:05

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

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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).

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I'm afraid I don't really see the connection; and as many other comments have noted, there is a well-established body of techniques to handle divergent series, which seem more relevant to the question at hand. – Yemon Choi Feb 4 '10 at 21:29
The question (specifically the part that said "Are there other ways to define such a thing") was, are there any ways to talk about and define sequence convergence, other than the standard definition from freshman calculus. My response is, look at the distribution of the values of the sequence and see if that distribution can be characterized somehow. For example, if the sequence is $a_n = (-1)^n$, then the distribution of its values is peaked at two points, +1 and -1. For a sequence $a_n = \sin(n)$, the distribution is different (covering almost uniformly much of $[-1, 1]$). – user3035 Feb 5 '10 at 1:32
The connection is that this gives a different and (depending on the application) potentially useful way to characterize a sequence and its limiting behavior. – user3035 Feb 5 '10 at 1:33
If I'm understanding it correctly, this technique subsumes some of the other limits mentioned, such as the one in the original question. Statistics of the distribution can give various limit notions. Seems like an interesting approach! – Matt Noonan Feb 5 '10 at 3:33
Thanks for this reply. Any additional information and any additional ideas are appreciated. – Rhubbarb Mar 9 '10 at 19:47

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

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...).

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you can take a look to these papers

Author explain in a simple fashion divergent series.

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