10/23/2008, 10:16 PM
Since there is more than one tetration, however, we do know a little bit about each (regular tetration, and natural tetration are 2 of them).
If you wanted to know more about natural tetration, here is an approximate power series:
\( {}^{x}e = \text{sexp}(x) = 1.0917(x+1) - 0.3244(x+1)^2 + 0.3498(x+1)^3 - 0.2308(x+1)^4 + \cdots \)
One of the problems with this method, though, is that the coefficients of the power series are not exact. You can find out more about natural tetration and its power series from this thread and this thread for the inverse function.
If you wanted to know more about regular tetration, then I suggest you read this thread, since it seems to be the first definition of it on this forum. Regular tetration is nice, because the coefficients are exact, but the problem is that it the power series is not an expansion of sexp(x), but an expansion of \( \exp^x(z) \) about z, which is a completely different power series. If we simply set z=1, then it doesn't always converge, so we are stuck with a different problem.
I hope that helps clarify a bit.
Andrew Robbins
If you wanted to know more about natural tetration, here is an approximate power series:
\( {}^{x}e = \text{sexp}(x) = 1.0917(x+1) - 0.3244(x+1)^2 + 0.3498(x+1)^3 - 0.2308(x+1)^4 + \cdots \)
One of the problems with this method, though, is that the coefficients of the power series are not exact. You can find out more about natural tetration and its power series from this thread and this thread for the inverse function.
If you wanted to know more about regular tetration, then I suggest you read this thread, since it seems to be the first definition of it on this forum. Regular tetration is nice, because the coefficients are exact, but the problem is that it the power series is not an expansion of sexp(x), but an expansion of \( \exp^x(z) \) about z, which is a completely different power series. If we simply set z=1, then it doesn't always converge, so we are stuck with a different problem.
I hope that helps clarify a bit.
Andrew Robbins