11/06/2007, 06:15 PM
Gottfried Wrote:I think, it has even the characteristic of a paradigm-change to move from the view as "appending exponents" to the view of "appending bases" - this seems to be much more important than the difficulties of introducing another notation. The question is: how do we embed our view and handling of tetration in the case of infinite height in the very basic and important concept of approximation via partial evaluation (partial sums).
"paradigm-change" is perhaps a bit exaggarated.
I didnt know of anybody who thinks of tetration as appending exponents. The right-bracketing rather forces one to use appending of bases, which is equivalent to \( {^nb}=f^{\circ n}(1) \) for \( f(x)=b^x \).
I also dont know what you mean by different partial evaluation, we already know that \( {^nb}=\{b,1\}^n \) in your notation. So (finite) tetration is just a particular case of appending bases. So the partial evaluations of \( {^\infty x} \) is exactly about appending bases. Or what do you mean by partial evaluation?