ca.analysis and odes - Are there Generalisations of a Limit (for Just-divergent Sequences)?

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.