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  "body": "\n## Statement\n\nLet $a_1, a_2, \\dots$ be nonnegative real numbers.\nThen\n$$\\sum_{n=1}^{\\infty} (a_1 a_2 \\cdots a_n)^{1/n} \\le e \\sum_{n=1}^{\\infty} a_n,$$\nwhere both [[series|0CVVKY7GY2]] take values in $[0, \\infty]$.\n\n## Remarks\n\nProof idea: let $c_k = (k + 1)^k / k^{k-1}$, so that $c_1 c_2 \\cdots c_n = (n + 1)^n$.\nBy the [[AM–GM inequality|0DRF0F3XC9]],\n$$(a_1 \\cdots a_n)^{1/n} = \\frac{(c_1 a_1 \\cdots c_n a_n)^{1/n}}{n + 1} \\le \\frac{1}{n(n+1)} \\sum_{k=1}^{n} c_k a_k.$$\nSumming over $n$ and using $\\sum_{n \\ge k} \\frac{1}{n(n+1)} = \\frac{1}{k}$ gives the bound $\\sum_k (1 + 1/k)^k a_k$, and $(1 + 1/k)^k \u003c e$.\n\nThe constant $e$ cannot be replaced by a smaller number.\nIf $0 \u003c \\sum a_n \u003c \\infty$, the inequality is strict.\n\n## Explanations\n\n### L1 human\n\nReplace each term of a series of nonnegative numbers by the geometric mean of the terms up to it.\nCarleman's inequality says the new series is at most $e$ times the old one.\nIn particular, if the old series converges, so does the new one.\n\n### L2 human\n\nCarleman's inequality is a limiting case of the discrete [[Hardy's inequality|S7A8YFGG8X]].\nApply Hardy's inequality to $a_n^{1/p}$ and let $p \\to \\infty$: the power means tend to geometric means, and $\\big(\\frac{p}{p-1}\\big)^p \\to e$.\nThe direct proof above shows how a well chosen weighting makes the AM–GM inequality sharp enough.\n",
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