{
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  "body": "\n## Statement\n\nLet $N$ be an integer and let $f \\colon [N, \\infty) \\to \\mathbb{R}$ be a decreasing function with $f(x) \\ge 0$ for all $x \\ge N$.\nThen the series $\\sum_{n=N}^{\\infty} f(n)$ converges if and only if the [[improper integral|0VCTBJMEJF]] $\\int_N^{\\infty} f(x)\\, dx$ converges.\n\n## Remarks\n\nA decreasing function is Riemann integrable on each bounded interval, so the integrals $\\int_N^b f(x)\\, dx$ exist.\nThe proof uses the [[comparison of a series and an integral|069HKV4X1E]]. When both converge,\n$$\\int_N^{\\infty} f(x)\\, dx \\le \\sum_{n=N}^{\\infty} f(n) \\le f(N) + \\int_N^{\\infty} f(x)\\, dx.$$\nWith $f(x) = 1/x^p$ it gives the [[p-series test|0ZNSSK8ZK1]].\nMathlib states the convergence direction for an antitone function, in `AntitoneOn.summable_of_integrableOn_Ioi`.\n\n## Explanations\n\n### L1 human\n\nDraw rectangles of width $1$ and height $f(n)$ under or over the graph of $f$.\nThe total area of the rectangles is the sum of the series, and it stays within $f(N)$ of the area under the graph.\n",
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