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  "body": "\n## Definition\n\nLet $(a_n)_{n \\ge 1}$ be a [[real sequence|0CTV783HYT]].\nFor each integer $n \\ge 1$, the $n$-th Cesàro mean of $(a_n)$ is the arithmetic mean of its first $n$ terms,\n$$\\sigma_n = \\frac{a_1 + a_2 + \\cdots + a_n}{n}.$$\n\n## Remarks\n\nThe sequence $(-1)^{n+1}$ diverges, but its Cesàro means are $1, 0, 1/3, 0, 1/5, \\dots$, and they tend to $0$.\nIf $(a_n)$ converges, the [[Cesàro means converge to the same limit|0RDRPBMNAC]].\nCesàro summation applies the Cesàro means to the partial sums of a series. A series is Cesàro summable with sum $A$ when the means of its partial sums tend to $A$.\n\n## Explanations\n\n### L1 human\n\nThe Cesàro mean is the running average of the sequence.\nAveraging smooths out oscillations. The sequence $1, 0, 1, 0, \\dots$ has no limit, but its running averages tend to $1/2$.\n",
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