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 Arithmetic in the height-parameter (sums, series) Gottfried Ultimate Fellow Posts: 757 Threads: 116 Joined: Aug 2007 02/04/2010, 05:08 PM (This post was last modified: 02/04/2010, 05:27 PM by Gottfried.) Hi - using the notation $x_1 = \exp_b^{^{oh_1}}(x_0)$ and $x_2 = \exp_b^{^{oh_2}}(x_1)$ we do arithmetic in the height (or "iteration") parameter like $x_2 = \exp_b^{^{oh_1+h_2}}(x_0)$ What about infinite series instead of a sum? If we have a sufficient method for continuous tetration, then, for instance we should get $x_2 = \exp_b^{^{o1/2+1/4+1/8+...}}(x_0) = \exp_b^{^{o1}}(x_0) = b^{x_0}$ For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge. So, with base b=sqrt(2) the following expression $y = \exp_b^{^{o1 + 2 + 4+ 8 +...}}(x_0)$ seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1. On the other hand, the analytical continuation for the geometric series with constant quotient q $g(q) = 1+q+q^2+ ...$ at q=2 gives $g(2) = 1/(1-2) = -1$ But -substitued this into the height-parameter- then we should also have $y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0$ where we see a contradiction. So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects). --- In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e: $y = e^{^{-1 -2 -4 -8 -16 - \dots }} = \frac1{e^1}*\frac1{e^2}*\frac1{e^4} * \dots = 0 \neq e^1$ Just another plot of meditations... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread Arithmetic in the height-parameter (sums, series) - by Gottfried - 02/04/2010, 05:08 PM RE: Arithmetic in the height-parameter (sums, series) - by bo198214 - 02/04/2010, 10:01 PM RE: Arithmetic in the height-parameter (sums, series) - by Gottfried - 02/05/2010, 10:23 AM Interpretation of summation techniques - by bo198214 - 02/05/2010, 12:31 PM RE: Interpretation of summation techniques - by Gottfried - 02/05/2010, 04:19 PM RE: Interpretation of summation techniques - by bo198214 - 02/05/2010, 06:45 PM RE: Interpretation of summation techniques - by Gottfried - 02/05/2010, 10:10 PM RE: Interpretation of summation techniques - by bo198214 - 02/06/2010, 12:52 AM

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