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  "body": "\n## Summary\n\nCesàro summation is the [[summability method|8X25Q4XYMN]] that assigns to a series the limit of the averages of its partial sums, when this limit exists.\nThese averages are the [[Cesàro means|08R2V9GZF7]] of the sequence of partial sums.\nEvery convergent series is Cesàro summable to its sum, and some divergent series, such as $1 - 1 + 1 - \\cdots$, become summable.\nWeighted versions of the averaging give the Cesàro methods $(C, \\alpha)$ of higher order.\nThe method of order $1$ is used for Fourier series in [[Fejér's theorem|TPJZCSXSVN]].\n\n## Explanations\n\n### L1 human\n\nTake the running totals of a series and then average them.\nFor $1 - 1 + 1 - 1 + \\cdots$, the running totals are $1, 0, 1, 0, \\dots$, which never settle down.\nTheir averages are $1, 1/2, 2/3, 1/2, 3/5, \\dots$, and these tend to $1/2$.\nSo Cesàro summation gives the value $1/2$ to this series.\nAveraging smooths out the oscillation while leaving the limit of a convergent series unchanged.\n",
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