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  "body": "\n## Definition\n\nLet $\\sum_{k \\ge 0} a_k$ be a [[series|0JH4MN4K1C]] of real or complex numbers with [[partial sums|0NB643K60Q]] $s_k = a_0 + \\dots + a_k$.\nLet $q \u003e 0$ be a real number and let $s$ be a number.\nThe series is *strongly Cesàro summable* with exponent $q$ to $s$ if\n$$\\lim_{n \\to \\infty} \\frac{1}{n + 1} \\sum_{k = 0}^{n} \\abs{s_k - s}^q = 0.$$\n\n## Remarks\n\nExample: a convergent series with sum $s$ is strongly Cesàro summable to $s$ for every $q \u003e 0$.\nIndeed $\\abs{s_k - s}^q \\to 0$, and the arithmetic means of a sequence tending to $0$ also tend to $0$.\n\nNon-example: Grandi's series $1 - 1 + 1 - \\cdots$ is [[Cesàro summable|N8ZGG7EW6J]] to $1/2$, but it is not strongly Cesàro summable with any exponent.\nIts partial sums alternate between $1$ and $0$.\nFor any $s$, one of $\\abs{1 - s}$ and $\\abs{s}$ is at least $1/2$.\nSo at least half of the terms $\\abs{s_k - s}^q$ are at least $2^{-q}$, and the means do not tend to $0$.\n\nFor $q \\ge 1$, the power mean inequality gives\n$$\\abs{\\frac{1}{n+1} \\sum_{k=0}^{n} (s_k - s)} \\le \\left( \\frac{1}{n+1} \\sum_{k=0}^{n} \\abs{s_k - s}^q \\right)^{1/q},$$\nso strong Cesàro summability to $s$ implies Cesàro summability to $s$.\n",
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