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?