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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $f_n \\colon [a, b] \\to \\mathbb{R}$ be differentiable for each $n \\ge 0$.\nAssume that $\\sum_{n} f_n(x_0)$ converges for some $x_0 \\in [a, b]$, and that $\\sum_{n} f_n'$ converges uniformly on $[a, b]$.\nThen $\\sum_{n} f_n$ converges uniformly on $[a, b]$, its sum is differentiable, and\n$$\\Big( \\sum_{n=0}^{\\infty} f_n \\Big)' = \\sum_{n=0}^{\\infty} f_n'.$$\n\n## Remarks\n\nApply [[the theorem on sequences|0CS42KEM5C]] to the partial sums.\nThe derivative series often converges uniformly by the M-test. Mathlib states that form as `hasDerivAt_tsum`.\nPower series satisfy the hypotheses on each closed interval inside the interval of convergence.\n\n## Explanations\n\n### L1 human\n\nWe may differentiate a series term by term, if the series of derivatives converges uniformly.\nThen the derivative of the sum is the sum of the derivatives.\n",
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  "name": "Term-by-term differentiation",
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