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  "body": "\n## Example\n\nThe infinite nested radical\n$$\\sqrt{1 + \\sqrt{1 + \\sqrt{1 + \\cdots}}}$$\nis defined as the limit of the sequence $x_1 = 1$, $x_{n+1} = \\sqrt{1 + x_n}$.\nThis is an [[infinite composition|B88PFYXWBK]] of the map $f(x) = \\sqrt{1 + x}$, since $x_n = f^{\\circ n}(0)$.\nIts value is the [[golden ratio|0SC8T7548D]] $\\varphi = \\frac{1 + \\sqrt5}{2}$.\n\nCheck: the sequence is increasing, since $x_2 = \\sqrt2 \u003e x_1$ and $f$ is increasing.\nIt is bounded above by $2$, since $x_n \u003c 2$ gives $x_{n+1} \u003c \\sqrt3 \u003c 2$.\nBy the [[monotone convergence theorem|0FM6N4R0VB]] it has a limit $L \\ge 1$, and continuity of $f$ gives $L = \\sqrt{1 + L}$.\nSo $L^2 - L - 1 = 0$, and the positive root is $L = \\varphi$.\n\nThe same method gives $\\sqrt{2 + \\sqrt{2 + \\cdots}} = 2$, from $L = \\sqrt{2 + L}$.\n\n## Explanations\n\n### L1 human\n\nAn infinite nested radical is a square root inside a square root, again and again.\nYou give it a meaning by stopping after $n$ roots and letting $n$ grow.\nThe partial values $1, 1.414, 1.554, 1.598, \\dots$ climb towards $1.618\\ldots$, the golden ratio.\n",
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