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  "body": "\n## Statement\n\nThe series $\\sum_{p} \\frac{1}{p}$, where $p$ runs over the [[prime numbers|0SN0MAEKYX]], diverges.\nThat is, the partial sums $\\sum_{p \\le x} \\frac{1}{p}$ are not bounded as $x \\to \\infty$.\n\n## Remarks\n\nThe result implies [[Euclid's theorem|0292EPPPT0]], since a finite sum is bounded.\nThe partial sums grow like $\\ln \\ln x$. More precisely, $\\sum_{p \\le x} \\frac{1}{p} - \\ln \\ln x$ has a finite limit as $x \\to \\infty$.\nIn contrast, the sum of $1 / n^2$ over all positive integers $n$ converges. So the primes are not as rare as the squares.\n\n## History\n\nEuler proved the result in 1737. Erdős gave an elementary proof by upper and lower estimates.\n\n## Explanations\n\n### L1 human\n\nThe sum $\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{7} + \\frac{1}{11} + \\cdots$ becomes larger than each number that you choose, if you take enough terms.\nIt grows very slowly: the terms for the primes up to $10^9$ still give a sum less than $4$.\n",
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