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  "body": "\n## Example\n\nLet $p$ be a real number. The [[improper integral|0VCTBJMEJF]] $\\int_1^\\infty x^p \\, dx$ converges if and only if $p \u003c -1$.\nIn this case,\n$$\\int_1^\\infty x^p \\, dx = -\\frac{1}{p + 1}.$$\nFor $p \\ne -1$, [[the power formula|0Q3HYFT9RQ]] gives $\\int_1^b x^p \\, dx = \\frac{b^{p+1} - 1}{p + 1}$. This has a finite limit as $b \\to \\infty$ exactly when $p + 1 \u003c 0$.\nFor $p = -1$, $\\int_1^b \\frac{dx}{x} = \\ln b$, [[which tends to infinity|0K1N1Y4EGZ]].\n\n## Remarks\n\nThis example parallels the [[$p$-series test|0ZNSSK8ZK1]] through the [[integral test|019QZ6RRVH]].\nMathlib states a form as `integrableOn_Ioi_rpow_iff`.\n\n## Explanations\n\n### L1 human\n\nThe function $x^p$ must decay faster than $1/x$ for the area over $[1, \\infty)$ to be finite.\nThe function $1/x$ itself is just too slow.\n",
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