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 Transseries, nest-series, and other exotic series representations for tetration kobi_78 Junior Fellow Posts: 11 Threads: 3 Joined: Dec 2009 12/14/2009, 02:11 PM (This post was last modified: 12/14/2009, 02:12 PM by kobi_78.) (12/14/2009, 01:33 PM)bo198214 Wrote: (12/08/2009, 10:32 AM)mike3 Wrote: So far, I've found 2 types of transseries representation such that if the series converges, the continuum sum does as well (proof given earlier here): I was a bit puzzled why $e^{e^x}$ converges via transseries while $e^{x^2}$ does not converge. Actually $e^{\frac{x^2}{2}}$ must also converge as we have $\frac{x^2}{2}, and indeed it also converges! But it seems that this already the limit, i.e. every series $e^{c x^2}$ for $c>\frac{1}{2}$ does not converge. I guess that this is the same for $e^{\frac{x^n}{n!}$, $n>1$, while Faulhaber always works for $n=1$, i.e. $e^{c x}$. Do you have any idea how to sum $e^{x^2}$? Hi, I've been reading on this forum for a while, and just now I've signed up. I have been exploring the sum operator, what you guys call "continuum sum" for a couple of years as a hobby. I think I have an idea how to sum $e^{x^2}$. Recall that $e^{x^2} = \lim_{n \to \infty} \left( 1 + \frac{x^2}{n} \right)^n$ Now calculate the polynomial sum of $\left( 1 + \frac{x^2}{n} \right)^n$ (using Faulhaber's formula). This seems to convergent to a nice function. « 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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