40-00-CesaroMean.tex

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\pmtitle{Ces\`aro mean}
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\pmtype{Definition}
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\begin{document}
\paragraph{Definition}
Let $\sequence{a_n}_{n=0}^\infty$ be a sequence of real (or possibly complex numbers).  
The {\bf Ces\`aro mean} of the sequence $\{a_n\}$ is the sequence $\{b_n\}_{n=0}^\infty$ 
with
\begin{equation}
b_n = \frac{1}{n+1} \sum_{i=0}^{n} a_i.
\end{equation}

\subsubsection{Properties}
\begin{enumerate}
\item
A key property of the Ces\`aro mean is that it has the same limit as the
original sequence (when this limit exists). In other words, if $\{a_n\}$ and 
$\{b_n\}$ are as above, and $a_n \to a$, then $b_n \to a$. 
In particular, if $\{a_n\}$ converges, then $\{b_n\}$ converges too.
\end{enumerate}
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