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 Transseries, nest-series, and other exotic series representations for tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/28/2009, 06:50 AM (This post was last modified: 11/28/2009, 07:42 AM by mike3.) Hmm. My hypothesis that the double-sum, or at least the exp-series, can only provide continuum sums of functions for "double-exponential type" or less seems to be wrong. Indeed, it seems that if an exp-series exists and converges for a function, then its continuum sum does too. Consider the triple-exponential function $f(x) = \exp^3(x) = e^{e^{e^x}}$. This can be expressed as an exp-series $e^{e^{e^x}} = \sum_{n=0}^{\infty} a_n e^{nx}$ where $a_n$ are the coefficients of the Taylor series of $e^{e^x}$ at $x = 0$ (MacLaurin series). Taking the continuum sum gives, \begin{align}\sum_{n=0}^{x-1} e^{e^{e^n}} &= a_0 x + \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1} \left(e^{nx} - 1\right) \\ &= \left(a_0 - \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1}\right) x + \sum_{n=1}^{\infty} \frac{a_n}{e^{n} - 1} e^{nx}\end{align} As the coefficients of both sums are smaller than those of the original sum (because $e^{n} - 1 > 0$ for all $n > 0$), if the original series converges at 0 and at the point $x$, so does this (see series comparison test). Since we have an expression for $e^{e^{e^x}}$, we have achieved its continuum sum and the proof (or disproof, insofar as my original hypothesis that such a series could not yield a continuum sum of something faster than a double exponential, but as a proof this proves even more, namely that any convergent exp-series' continuum sum also converges, unlike the case with Taylor series summed via direct application of Faulhaber's formula) is complete. As exp-series look to be a special case of nested series, this suggests even 2 layers of nesting may be able to represent tetration and continuum-sum it, though there's still no proof for that. The special case of exp-series themselves do not appear useful for doing tetration with Ansus' formula, however, for two reasons: any function constructed with them is $2\pi i$-periodic, yet tetration seems not to be given pretty much every "good" extension there is -- though they may be able to express tetration for the base $b = e^{e^{1-e}}$ whose regular tetration is periodic with the required period (and so the exp-series can be recovered via the Fourier series), but a single base isn't very useful. And, we can't even represent $f(x) = x$ as an exp-series, thus we can't even continuum-sum one exp-series to another exp-series so this is not very useful insofar as trying to iteratively apply Ansus' formula to generate tetrationals goes! « Next Oldest | Next Newest »

 Messages In This Thread Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/26/2009, 09:46 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/26/2009, 03:57 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/26/2009, 04:42 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/29/2009, 09:09 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/29/2009, 09:38 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 12/01/2009, 02:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 12/01/2009, 09:08 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by tommy1729 - 12/01/2009, 10:22 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/27/2009, 01:29 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by andydude - 11/28/2009, 04:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:36 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:50 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by kobi_78 - 12/14/2009, 07:17 PM

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