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

a_{1}, a_{2}, a_{3}, a_{4}, ...

the accumulated mean sequence would be

a_{1}, (a_{1}+a_{2})/2, (a_{1}+a_{2}+a_{3})/3, (a_{1}+a_{2}+a_{3}+a_{4})/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.