Cesàro summability is a generalized convergence criterion for infinite series. We say that a series is Cesàro summable if the Cesàro means of the partial sums converge to some limit . To be more precise, letting
denote the partial sum, we say that Cesàro converges to a limit , if
Cesàro summability is a generalization of the usual definition of the limit of an infinite series.
in the usual sense that as . Then, the series in question Cesàro converges to the same limit.
The sequence of partial sums does not converge. The Cesàro means, namely
do converge, with as the limit. Hence the series in question is Cesàro summable.
Theorem 2 (Frobenius)
A series that is Cesàro summable is also Abel summable. To be more precise, suppose that
|Date of creation||2013-03-22 13:07:01|
|Last modified on||2013-03-22 13:07:01|
|Last modified by||rmilson (146)|