(

Calculus:)

A

weaker "

limit-like" concept, related to

summability (in particular

Cesaro summability).

Let x_{1},x_{2},... be a sequence (in **R** or in some other normed space). Let

X_{n} = (x_{1}+...+x_{n})/n

be the sequence of

averages of the first n

elements. If

L = lim_{n->∞} X_{n}

exists, we say that the original sequence x

_{1},x

_{2},...

*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=lim
_{n->∞}x_{n} 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 x
_{n}'s: if we change the first n elements of the sequence, x'_{1}, ..., x'_{n}, x_{n+1}, x_{n+2}, ... has the same Cesaro limit as x_{1}, x_{2}, ... . "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 x
_{1}, x_{2}, ... (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).