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A weaker "limit-like" concept, related to summability (in particular Cesaro summability).

Let x1,x2,... be a sequence (in R or in some other normed space). Let

Xn = (x1+...+xn)/n
be the sequence of averages of the first n elements. If
L = limn->∞ Xn
exists, we say that the original sequence x1,x2,... has Cesaro limit L.

For example, the sequence 0,1,0,1,... (which has no limit!) has Cesaro limit 1/2. This is particularly satisfying, as it seems like the right number.

Cesaro limits are employed when analysing sequences which do not converge (in the sense of having a limit). They are a weaker concept than limits (but stronger than Banach limits):

  • The Cesaro limit is a linear functional on the space of sequences.
  • If the limit l=limn->∞xn exists, then the Cesaro limit L also exists, and L=l.
  • The existence and value of the Cesaro limit L is independent of any finite number of xn's: if we change the first n elements of the sequence, x'1, ..., x'n, xn+1, xn+2, ... has the same Cesaro limit as x1, x2, ... . "Real" limits have the same property, of course.
  • The Cesaro limit L exists for many sequences which have no limit. For instance, the sequence 0,1,0,1,0,1,... above.
  • The Banach limit of a bounded sequence x1, x2, ... (which always exists) is equal to the Cesaro limit, if the Cesaro limit exists.

Cesaro limits, and especially the related concept of Cesaro summability, is extensively used in analysis. For instance, at points where a Fourier series fails to converge, the Cesaro limit of the sum often will converge (and have mathematically pleasing (and useful) properties).

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