Cesàro summation

Summary

Cesàro summation is the summability method that assigns to a series the limit of the averages of its partial sums, when this limit exists. These averages are the Cesàro means of the sequence of partial sums. Every convergent series is Cesàro summable to its sum, and some divergent series, such as $1 - 1 + 1 - \cdots$, become summable. Weighted versions of the averaging give the Cesàro methods $(C, \alpha)$ of higher order. The method of order $1$ is used for Fourier series in Fejér's theorem.