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  "body": "\n## Statement\n\nLet $(x_n)_{n \\ge 1}$ be a real sequence such that the series $\\sum_{n=1}^{\\infty} x_n$ [[converges|0CVVKY7GY2]].\nLet $(b_n)_{n \\ge 1}$ be a [[monotone|0KN22AAW5R]] sequence of real numbers with $0 \u003c b_1 \\le b_2 \\le \\cdots$ and $b_n \\to \\infty$.\nThen\n$$\\lim_{n \\to \\infty} \\frac{1}{b_n} \\sum_{k=1}^{n} b_k x_k = 0.$$\n\n## Remarks\n\nLet $r_k = \\sum_{j \u003e k} x_j$ be the tails of the series, so $x_k = r_{k-1} - r_k$ with $r_k \\to 0$.\nSummation by parts writes $\\frac{1}{b_n} \\sum_{k \\le n} b_k x_k$ as $b_1 r_0 / b_n - r_n$ plus a weighted average of $r_1, \\ldots, r_{n-1}$ with weights $(b_{k+1} - b_k)/b_n$.\nThese weights are nonnegative, their sum is at most $1$, and each fixed weight tends to $0$, so the average tends to $0$ because $r_k \\to 0$.\nExample: with $b_n = n$, if $\\sum y_n / n$ converges, then $(y_1 + \\cdots + y_n)/n \\to 0$.\n\n## Explanations\n\n### L1 human\n\nIf a series converges, its terms become small on average.\nKronecker's lemma makes this precise: even after multiplying the terms by growing weights $b_k$, the weighted sum grows more slowly than $b_n$.\n\n### L2 human\n\nThe main use is in probability.\nTo prove a strong law of large numbers, one shows that $\\sum X_n / n$ converges almost surely, and Kronecker's lemma with $b_n = n$ then gives $(X_1 + \\cdots + X_n)/n \\to 0$.\nThe proof is a summation by parts followed by an averaging argument, as in the [[Stolz–Cesàro theorem|0Q3AKZV74D]].\n",
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