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  "body": "\n## Example\n\nThe *Kempner series* is the sum of $1/n$ over the positive integers $n$ whose decimal expansion has no digit $9$:\n$$\\sum_{\\substack{n \\ge 1 \\\\ \\text{no digit } 9}} \\frac{1}{n} = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{8} + \\frac{1}{10} + \\cdots + \\frac{1}{18} + \\frac{1}{20} + \\cdots.$$\nIt is a [[convergent series|0AF7X08PCR]], and its sum is less than $80$.\n\nCheck: a positive integer with exactly $k$ digits and no digit $9$ has a first digit in $\\set{1, \\ldots, 8}$ and $k - 1$ further digits in $\\set{0, \\ldots, 8}$.\nSo there are $8 \\cdot 9^{k-1}$ such integers, and each is at least $10^{k-1}$.\nTheir reciprocals add up to at most $8 \\cdot 9^{k-1} / 10^{k-1} = 8 \\cdot (9/10)^{k-1}$.\nSumming over $k \\ge 1$ gives the bound $8 / (1 - 9/10) = 80$.\nThe terms are positive, so the partial sums are increasing and bounded, and the series converges.\n\n## Remarks\n\nThe true sum is about $22.92$.\nThis contrasts with the [[harmonic series|0CXTMAY5PX]], which [[diverges|0E9M24KS9J]].\nThe same argument works for any digit and any base: removing every integer that contains a fixed digit string leaves a convergent series.\nThe series is named after A. J. Kempner, who proved its convergence in 1914.\nMost large integers contain a $9$: the proportion of $k$-digit integers without one is $(8/9)(9/10)^{k-1}$, which tends to $0$.\n\n## Explanations\n\n### L1 human\n\nThe harmonic series $1 + 1/2 + 1/3 + \\cdots$ grows without bound.\nThrow away every term whose denominator has a $9$ somewhere in it.\nThis looks like a small change, but almost all large numbers contain a $9$, so most terms disappear, and what remains adds up to a finite number, about $22.92$.\n",
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