02/04/2010, 10:01 PM
(02/04/2010, 05:08 PM)Gottfried Wrote: 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
\( y = 2 = \exp_b^{^{o -1 }}(x_0) = \log_b(x_0) = \log_b(1) = 0 \)
Well, the equality 1 + 2 + 4 + 8 + ... = -1, is very fragile if you want to take it as a serious argument. It doesnt surprise me that you dont get the result you expect.
