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

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

Gottfried Helms, Kassel