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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence and let $C = \\limsup_{n \\to \\infty} |a_n|^{1/n}$, the [[limit superior|0P8BA7T7S5]], with the value $+\\infty$ allowed.\nIf $C \u003c 1$, then the series $\\sum a_n$ converges [[absolutely|0W3K3QPEWT]].\nIf $C \u003e 1$, then the series $\\sum a_n$ diverges.\n\n## Remarks\n\nIf $C \u003c 1$, choose $q$ with $C \u003c q \u003c 1$. Then $|a_n| \\le q^n$ for large $n$, and the [[comparison test|04Z5G0THN6]] with a [[geometric series|01X5CVV1Q5]] applies.\nIf $C \u003e 1$, then $|a_n| \u003e 1$ for infinitely many $n$, so the terms do not tend to $0$.\nFor $C = 1$ the test gives no answer.\nThe root test is stronger than the [[ratio test|0QNY6ZBXPN]]: when the ratio limit exists, the root limit exists and has the same value.\n\n## Explanations\n\n### L1 human\n\nThe $n$-th root of $|a_n|$ measures how fast the terms shrink, compared with a geometric sequence $q^n$.\nIf this rate is below $1$, the terms shrink like a convergent geometric series.\n",
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