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.



Are there Generalisations of a Limit (for Just-divergent Sequences)?
Another common technique is [Abel summation](http://en.wikipedia.org/wiki/Divergent_series#Abel_summation), which works a little better than Cesaro summation. [Zeta regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization) is also important in physics.


You might enjoy reading [these posts at The Everything Seminar](http://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/) and [this column from John Baez](http://math.ucr.edu/home/baez/week126.html).


