In mathematical analysis, CesÃÂ ro summation assigns values to some infinite sums that are not necessarily convergent in the usual sense. The CesÃÂ ro sum, also known as the CesÃÂ ro mean or CesÃÂ ro limit, is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
This special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859âÂÂ1906).
The term summation can be misleading, as some statements and proofs regarding Cesàro summation can be said to implicate the EilenbergâÂÂMazur swindle. For example, it is commonly applied to Grandi's series with the conclusion that the sum of that series is 1/2.
Let be a sequence, and let
be its th partial sum.
The sequence is called CesÃÂ ro summable, with CesÃÂ ro sum , if, as tends to infinity, the arithmetic mean of its first n partial sums tends to :
The value of the resulting limit is called the CesÃÂ ro sum of the series If this series is convergent, then it is CesÃÂ ro summable and its CesÃÂ ro sum is the usual sum.
Let for . That is, is the sequence
Let denote the series
The series is known as Grandi's series.
Let denote the sequence of partial sums of :
This sequence of partial sums does not converge, so the series is divergent. However, CesÃÂ ro summable. Let be the sequence of arithmetic means of the first partial sums:
Then
and therefore, the CesÃÂ ro sum of the series is .
As another example, let for . That is, is the sequence
Let now denote the series
Then the sequence of partial sums is
Since the sequence of partial sums grows without bound, the series diverges to infinity. The sequence of means of partial sums of G is
This sequence diverges to infinity as well, so is CesÃÂ ro summable. In fact, for the series of any sequence which diverges to (positive or negative) infinity, the CesÃÂ ro method also leads to the series of a sequence that diverges likewise, and hence such a series is not CesÃÂ ro summable.
In 1890, Ernesto CesÃÂ ro stated a broader family of summation methods which have since been called for non-negative integers . The method is just ordinary summation, and is CesÃÂ ro summation as described above.
The higher-order methods can be described as follows: given a series , define the quantities
(where the upper indices do not denote exponents) and define to be for the series . Then the sum of is denoted by and has the value
if it exists . This description represents an -times iterated application of the initial summation method and can be restated as
Even more generally, for , let be implicitly given by the coefficients of the series
and as above. In particular, are the binomial coefficients of power . Then the sum of is defined as above.
If has a sum, then it also has a sum for every , and the sums agree; furthermore we have if (see little- notation).
Let . The integral is summable if
exists and is finite . The value of this limit, should it exist, is the sum of the integral. Analogously to the case of the sum of a series, if , the result is convergence of the improper integral. In the case , convergence is equivalent to the existence of the limit
which is the limit of means of the partial integrals.
As is the case with series, if an integral is summable for some value of , then it is also summable for all , and the value of the resulting limit is the same.