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tiny q: superroots of real numbers x>e - Gottfried - 02/02/2009
Hi folks - I've just asked this question in news:sci.math; it is a tiny question and possibly answered anywhere here around ( I didn't follow the superroot-discussion intensely) so maybe we have a link already... Ok, let's go: Let's define the n'th iterative root ("srt") via Code: `f(x,1) = x f(x,2) = x^x f(x,3) = x^(x^x) f(x,k) = ...` Code: `srt(y,3) = x --> f(x,3) = y` Code: `srt(3,1) , srt(3,2), srt(3,3),..., srt(3,k),... (for k=1 ... inf )` Then: what is x in Code: `x = lim {k->inf} srt(3,k)` The sequence decreases from 3 down to e^(1/e) + eps but I think, it cannot fall below. Code: `k x=srt(3,k)` 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)? <urrks> Gottfried RE: tiny q: superroots of real numbers x>e - bo198214 - 02/02/2009
Gottfried Wrote:On the other hand, it should arrive at 3^(1/3)... Indeed a very interesting observation, Gottfried. You only arrive at the expected value if it is , i.e. only if . This is because for , where and . For , for example , is always for each . Suppose otherwise then would , for while . RE: tiny q: superroots of real numbers x>e - Gottfried - 02/03/2009
Hi Henryk - It's late, I can't comment/proceed at the moment, let's see tomorrow. Here are two plots to illustrate the beginning of the trajectory, anyway. [attachment=436] [attachment=438] Nächtle... ;-) [update] pic changed [/update] Gottfried RE: tiny q: superroots of real numbers x>e - bo198214 - 02/03/2009
To be clear: I think its sure that for and for I would guess: this also corresponds to your pictures. RE: tiny q: superroots of real numbers x>e - Gottfried - 02/03/2009
Yepp, so we have the interesting property, that we have two numbers: a proper limit (e^(1/e)) for the sequence of srt of increasing order and x^(1/x) as value for "the immediate" evaluation of the infinite expression. Hmm - surely this should be formulated more smoothly. Can we then say, that the infinite iterative root for y>e has two values? ... so many questions... Gottfried RE: tiny q: superroots of real numbers x>e - bo198214 - 02/03/2009
Gottfried Wrote:Can we then say, that the infinite iterative root for y>e has two values? No, we have two cases and for each case one limit. |