Tuesday, 4 January 2011

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.

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