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  "body": "\n## Summary\n\nHölder summation is a [[summability method|8X25Q4XYMN]] that averages the partial sums of a series, then averages these averages.\nIt repeats the averaging a fixed number $k$ of times.\nFor $k = 1$ it is [[Cesàro summation|N8ZGG7EW6J]] of order $1$.\nFor each $k$, the Hölder method of order $k$ sums the same series to the same values as the Cesàro method of order $k$.\nRepeated averaging can tame series whose partial sums oscillate with growing size.\n\n## Explanations\n\n### L1 human\n\nOne round of averaging turns the running totals $1, 0, 1, 0, \\dots$ of $1 - 1 + 1 - \\cdots$ into a sequence that tends to $1/2$.\nSome series need more rounds.\nFor $1 - 2 + 3 - 4 + \\cdots$, one round of averaging gives a sequence that still jumps between about $1/2$ and $0$.\nA second round averages this out and gives $1/4$.\nHölder summation of order $k$ simply means $k$ rounds of averaging.\n",
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