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 tiny q: superroots of real numbers x>e bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/02/2009, 10:21 PM (This post was last modified: 02/02/2009, 10:31 PM by bo198214.) Gottfried Wrote:On the other hand, it should arrive at 3^(1/3)... Do I actually overlook something and the sequence can indeed cross e^(1/e)? Indeed a very interesting observation, Gottfried. You only arrive at the expected value if it is $, i.e. $\lim_{k\to\infty} \text{srt}(x,k)=\sqrt[x]{x}$ only if $1\le x\le e$. This is because $1\le {^\infty}b\le e$ for $1\le b\le e^{1/e}$, where $x={^\infty}b$ and $b=\sqrt[x]{x}$. For $x>e$, for example $x=3$, is always $\text{srt}(x,k) > e^{1/e}$ for each $k$. Suppose otherwise $\text{srt}(x,k)\le e^{1/e}=:y$ then would $x\le {^k}y$, for $y\le e^{1/e}$ while ${^k}y\le e$. « Next Oldest | Next Newest »

 Messages In This Thread tiny q: superroots of real numbers x>e - by Gottfried - 02/02/2009, 06:54 PM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/02/2009, 10:21 PM RE: tiny q: superroots of real numbers x>e - by Gottfried - 02/03/2009, 12:15 AM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/03/2009, 10:01 AM RE: tiny q: superroots of real numbers x>e - by Gottfried - 02/03/2009, 11:36 AM RE: tiny q: superroots of real numbers x>e - by bo198214 - 02/03/2009, 12:46 PM

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