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Cesàro summability
Cesàro summability is a generalized convergence criterion for infinite series. We say that a series $\sum_{n=0}^\infty a_n$ is Cesàro summable if the Cesàro means of the partial sums converge to some limit $L$ . To be more precise, letting $$s_N=\sum_{n=0}^N a_n$$ denote the $N\supth$ partial sum, we say that $\sum_{n=0}^\infty a_n$ Cesàro converges to a limit $L$ , if $$\frac{1}{N+1}(s_0+\ldots+s_N) \rightarrow L \quad\text{as}\quad N\rightarrow\infty.$$
Cesàro summability is a generalization of the usual definition of the limit of an infinite series.
Proposition 1 Suppose that $$\sum_{n=0}^\infty a_n = L,$$ in the usual sense that $s_N\rightarrow L$ as $N\rightarrow\infty$ . Then, the series in question Cesàro converges to the same limit.
The converse, however is false. The standard example of a divergent series, that is nonetheless Cesàro summable is $$\sum_{n=0}^\infty (-1)^n.$$ The sequence of partial sums $1,0,1,0,\ldots$ does not converge. The Cesàro means, namely $$\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6}, \ldots$$ do converge, with $1/2$ as the limit. Hence the series in question is Cesàro summable.
There is also a relation between Cesàro summability and Abel summability 1.
Theorem 2 (Frobenius) A series that is Cesàro summable is also Abel summable. To be more precise, suppose that $$\frac{1}{N+1}(s_0+\ldots+s_N) \rightarrow L \quad\text{as}\quad N\rightarrow\infty.$$ Then, $$f(r) = \sum_{n=0}^\infty a_n r^n \rightarrow L \quad\text{as}\quad r\rightarrow 1^{-}$$ as well.
Footnotes
- ... summability1
- This and similar results are often called Abelian theorems.
Cesàro summability is owned by Robert Milson.
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