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 New tetration method based on continuum sum and exp-series tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 04/20/2010, 03:26 PM (04/20/2010, 02:48 AM)mike3 Wrote: Hi. We have $f(z) = \sum_{n=0}^{\infty} a_n b^{nz}$ or, even better $f(z) = \sum_{n=-\infty}^{\infty} a_n b^{nz}$. Then, $\sum_{n=0}^{z-1} f(n) = \sum_{n=-\infty}^{\infty} a_n \frac{b^{nz} - 1}{b^n - 1}$ with the term at $n = 0$ interpreted as $a_0 z$, so, $\sum_{n=0}^{z-1} f(n) = \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1}\right) + a_0 z + \left(\sum_{n=-\infty}^{-1} \frac{a_n}{b^n - 1} b^{nz}\right) + \left(\sum_{n=1}^{\infty} \frac{a_n}{b^n - 1} b^{nz}\right)$. Thus it seems this continuum sum is recovering all the expected sums and extensions. So the question comes up: what happens if we use it on Tetration, to sum up Ansus' continuum sum formula $\log_b\left(\frac{\mathrm{tet}'_b(z)}{\mathrm{tet}'_b(0) \log(b)^z}\right) = \sum_{n=0}^{z-1} \mathrm{tet}_b(n)$ ? I don't yet have a really fast and efficient numerical program ready to go, but the idea behind the algorithm I'm using and the current code I can post in the Computation forum if you'd like. but you dont have the coefficients of ansus tet_b , so no taylor series ? and tet_b is not periodic either ? yes plz explain how you compute it ! regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread New tetration method based on continuum sum and exp-series - by mike3 - 04/20/2010, 02:48 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/20/2010, 10:40 AM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/20/2010, 09:10 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 10:20 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/22/2010, 10:35 AM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 10:57 AM RE: New tetration method based on continuum sum and exp-series - by bo198214 - 04/22/2010, 12:25 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/22/2010, 08:52 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/22/2010, 09:36 PM RE: New tetration method based on continuum sum and exp-series - by mike3 - 04/30/2010, 04:36 AM RE: New tetration method based on continuum sum and exp-series - by andydude - 05/02/2010, 09:27 AM RE: New tetration method based on continuum sum and exp-series - by andydude - 05/02/2010, 09:58 AM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 03:26 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 07:45 PM RE: New tetration method based on continuum sum and exp-series - by tommy1729 - 04/20/2010, 08:12 PM

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