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  "body": "\n## Summary\n\nStrong Cesàro summability asks that the partial sums of a series stay close to a value $s$ on average, measured by the mean of $\\abs{s_k - s}^q$.\nOrdinary [[Cesàro summation|N8ZGG7EW6J]] only asks that the mean of $s_k - s$ tends to $0$, which allows positive and negative deviations to cancel.\nFor exponent $q \\ge 1$, strong Cesàro summability implies Cesàro summability to the same value.\nIt is used in the theory of Fourier series and is closely tied to [[statistical convergence|ZW6EHSMANH]].\n\n## Explanations\n\n### L1 human\n\nGrandi's series $1 - 1 + 1 - \\cdots$ has partial sums $1, 0, 1, 0, \\dots$, whose average tends to $1/2$.\nBut each partial sum is at distance $1/2$ from $1/2$, so the partial sums are never close to the value.\nStrong summability forbids this: the average distance from the value must itself tend to $0$.\nSo the partial sums must be near $s$ most of the time, not only in balance.\n",
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