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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $f_n \\colon [a, b] \\to \\mathbb{R}$ be Riemann integrable for each $n \\ge 0$.\nIf $\\sum_{n} f_n$ converges uniformly on $[a, b]$, then its sum is Riemann integrable, and\n$$\\int_a^b \\sum_{n=0}^{\\infty} f_n(x) \\, dx = \\sum_{n=0}^{\\infty} \\int_a^b f_n(x) \\, dx.$$\n\n## Remarks\n\nApply [[the theorem on sequences|0BWGW68B66]] to the partial sums.\nFor example, integrating $\\frac{1}{1 + t} = \\sum (-t)^n$ term by term over $[0, x]$, with $|x| \u003c 1$, gives the Mercator series for $\\ln(1 + x)$.\nMathlib states a version with summable sup norms as `intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm`.\n\n## Explanations\n\n### L1 human\n\nWe may integrate a uniformly convergent series term by term.\nThe integral of the sum is the sum of the integrals.\n",
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