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  "body": "\n## Definition\n\nLet $S$ be a set and let $f_n \\colon S \\to \\mathbb{R}$ be bounded functions for $n \\ge 0$.\nThe [[series of functions|0MRE6XRE5V]] $\\sum_{n=0}^{\\infty} f_n$ converges normally on $S$ if\n$$\\sum_{n=0}^{\\infty} \\sup_{x \\in S} |f_n(x)| \u003c \\infty.$$\n\n## Remarks\n\nBy the [[Weierstrass M-test|0NX5NWAFC9]], a normally convergent series converges absolutely and uniformly on $S$.\nThe converse fails. For constant functions $f_n = (-1)^n / (n + 1)$, the series converges uniformly but not normally.\nRené Baire introduced the concept in 1908.\n\n## Explanations\n\n### L1 human\n\nNormal convergence means that the sizes of the functions, measured by their suprema, form a convergent series.\nIt is a simple condition that is stronger than uniform convergence.\n",
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