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  "body": "\n## Definition\n\nLet $\\sum_{n \\ge 0} a_n$ be a series of complex numbers with [[partial sums|0NB643K60Q]] $s_n = a_0 + \\cdots + a_n$.\nPut $H^0_n = s_n$, and for each integer $k \\ge 1$ let $H^k_n$ be the [[Cesàro mean|0P0DJHBGBB]] of the sequence $(H^{k-1}_n)$:\n$$H^k_n = \\frac{H^{k-1}_0 + H^{k-1}_1 + \\cdots + H^{k-1}_n}{n + 1}.$$\nThe series is *Hölder summable of order $k$*, or $(H, k)$ summable, to $s$ if $H^k_n \\to s$ as $n \\to \\infty$.\n\n## Remarks\n\nThe method $(H, 0)$ is ordinary convergence, and $(H, 1)$ is Cesàro summation of order $1$.\n\nExample: for $1 - 2 + 3 - 4 + \\cdots$ the partial sums are $1, -1, 2, -2, 3, \\dots$.\nThen $H^1_{2m} = \\frac{m+1}{2m+1}$ and $H^1_{2m+1} = 0$.\nSo $H^1_n$ is the sequence $\\frac12, 0, \\frac12, 0, \\dots$ up to terms that tend to $0$, and its averages $H^2_n$ tend to $\\frac14$.\nThe series is $(H, 2)$ summable to $1/4$.\n\nNon-example: the same series is not $(H, 1)$ summable, since $H^1_n$ has the two limit points $1/2$ and $0$.\n\nBy a theorem of Knopp and Schur, for each $k$ a series is $(H, k)$ summable to $s$ if and only if it is Cesàro $(C, k)$ summable to $s$.\n",
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