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  "body": "\n## Statement\n\nLet $S$ be a set and let $f_n \\colon S \\to \\mathbb{R}$ for $n \\ge 0$.\nAssume that there are real numbers $M_n \\ge 0$ with $|f_n(x)| \\le M_n$ for all $n$ and all $x \\in S$, and $\\sum_{n=0}^{\\infty} M_n \u003c \\infty$.\nThen the [[series of functions|0MRE6XRE5V]] $\\sum_{n=0}^{\\infty} f_n$ converges absolutely and uniformly on $S$.\n\n## Remarks\n\nAbsolute convergence at each $x$ follows from the [[comparison test|04Z5G0THN6]]. For $m \u003c n$, $|s_n(x) - s_m(x)| \\le \\sum_{k=m+1}^{n} M_k$. So the partial sums are [[uniformly Cauchy|01XAY17QA3]].\nFor example, $\\sum \\frac{\\sin(n x)}{n^2}$ converges uniformly on $\\mathbb{R}$, with $M_n = 1/n^2$.\nMathlib states it as `tendstoUniformlyOn_tsum`.\n\n## Explanations\n\n### L1 human\n\nIf each function is bounded by a number, and these numbers have a finite sum, then the series of functions converges uniformly.\nThe bounds do not depend on $x$, so the convergence is the same everywhere.\n",
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  "name": "Weierstrass M-test",
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