A Cesàro sum is a limit obtained by averaging the partial sums of a series.
Let 
be its partial
sums, and define their arithmetic means
 |
(1)
|
If
 |
(2)
|
then the series is Cesàro summable to 
, and the method of assigning this value is called Cesàro
summation. Every convergent series is Cesàro
summable to its ordinary sum, while some divergent
series are Cesàro summable. For example, 
has Cesàro sum 
.
This is the order-1 method, denoted 
. For a positive integer 
, the higher-order means are
 |
(3)
|
where the weights are binomial coefficients (Hardy 1949, p. 96). If 
,
the series is called Cesàro summable of order 
to 
. For integer 
, these means differ from repeatedly taking arithmetic
means, but give the same summability and sum as Hölder
regularization of order 
(Hardy 1949, p. 103).
For 
, the series 
is not summable 
because the even and odd subsequences of 
tend to 
and 0. Its order-2 means are
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is a nonnegative
integer. Both tend to 
,
so the series is summable 
. Every series summable 
for a positive
integer 
has an Abel-regularized sum equal to its
sum (Hardy
1949, p. 108).
