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  "body": "\n## Example\n\nFor $x \u003e 0$, the infinite power tower $x^{x^{x^{\\cdots}}}$ is defined as the limit of $t_1 = x$, $t_{n+1} = x^{t_n}$.\nIt is an [[infinite composition|B88PFYXWBK]] of the map $t \\mapsto x^t$.\nTake $x = \\sqrt2$; the tower converges to $2$.\n\nCheck: $t_1 = \\sqrt2 \u003c 2$, and if $t_n \u003c 2$ then $t_{n+1} = \\sqrt2^{\\,t_n} \u003c \\sqrt2^{\\,2} = 2$.\nSince $t \\mapsto \\sqrt2^{\\,t}$ is increasing and $t_2 = \\sqrt2^{\\sqrt2} \u003e \\sqrt2 = t_1$, the sequence is increasing.\nBy the [[monotone convergence theorem|0FM6N4R0VB]] it has a limit $L \\le 2$ with $\\sqrt2^{\\,L} = L$.\nThe equation $\\sqrt2^{\\,L} = L$ has the two solutions $L = 2$ and $L = 4$, and $L \\le 2$ gives $L = 2$.\n\nA classical result, going back to Euler, is that the tower converges exactly for $e^{-e} \\le x \\le e^{1/e}$.\nFor example, $x = 2 \u003e e^{1/e} \\approx 1.444$ gives $2, 4, 16, 65536, \\dots$, which tends to infinity.\n\n## Explanations\n\n### L1 human\n\nA power tower stacks the same number in the exponent again and again.\nFor $\\sqrt2$ the values creep up towards $2$ and never pass it, because $\\sqrt2$ raised to the power $2$ is again $2$.\nFor larger bases the tower explodes.\n",
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