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Arithmetic in the height-parameter (sums, series)
Hi -

using the notation and we do arithmetic in the height (or "iteration") parameter like

What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get

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

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 at q=2 gives

But -substitued this into the height-parameter- then we should also have

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:

Just another plot of meditations... <phew>

Gottfried Helms, Kassel

Messages In This Thread
Arithmetic in the height-parameter (sums, series) - by Gottfried - 02/04/2010, 05:08 PM

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