Ces\`aro summabilityCesaroSummability2002-10-29 10:45:322002-10-29 13:15:34Definition\usepackage{amsmath}
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\newtheorem{theorem}[proposition]{Theorem}Ces\`aro summability is a generalized convergence criterion for infinite
series. We say that a series $\sum_{n=0}^\infty a_n$ is
Ces\`aro summable if the Ces\`aro 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 $\sum_{n=0}^\infty a_n$
Ces\`aro converges to a limit $L$, if
$$\frac{1}{N+1}(s_0+\ldots+s_N) \rightarrow L \quad\text{as}\quad N\rightarrow\infty.$$
Ces\`aro summability is a generalization of the usual definition of the
limit of an infinite series.
\begin{proposition}
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 Ces\`aro converges to the same
limit.
\end{proposition}
The converse, however is false. The standard example of a divergent
series, that is nonetheless Ces\`aro summable is
$$\sum_{n=0}^\infty (-1)^n.$$
The sequence of partial sums
$1,0,1,0,\ldots$ does not converge. The Ces\`aro 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 Ces\`aro summable.
There is also a relation between Ces\`aro summability and Abel
summability\footnote{This and similar results are often called Abelian
theorems.}.
\begin{theorem}[Frobenius]
A series that is Ces\`aro 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.
\end{theorem}