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Let $\sequence{a_n}_{n=0}^\infty$ be a sequence of real (or possibly complex numbers). The Cesàro 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}
- A key property of the Cesàro 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.
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"Cesàro mean" is owned by mathcam. [ full author list (3) | owner history (2) ]
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Cross-references: converges, limit, property, complex numbers, real, sequence
There are 4 references to this entry.
This is version 8 of Cesàro mean, born on 2002-02-27, modified 2004-07-29.
Object id is 2725, canonical name is CesaroMean.
Accessed 8116 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) | | | 40G05 (Sequences, series, summability :: Special methods of summability :: Cesàro, Euler, Nörlund and Hausdorff methods) |
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Pending Errata and Addenda
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