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Math.Stackexchange.com question on extending tetration - Daniel - 03/28/2021

Howdy,
Check out a question on extending tetration. I'm inviting folks to critic my answers or to provide your own.
Daniel


RE: Math.Stackexchange.com question on extending tetration - JmsNxn - 03/29/2021

I liked that post. Cool graphics, it's nice to see some love for the standard iteration--and a cool Taylor Series for it.


RE: Math.Stackexchange.com question on extending tetration - nuninho1980 - 03/30/2021

(03/28/2021, 06:26 AM)Daniel Wrote: Howdy,
Check out a question on extending tetration. I'm inviting folks to critic my answers or to provide your own.
Daniel

I use Maple 2020. Wink

base^^x = f(x) = lim n -> infinity (log_base[n](1 - ln(W(-ln(base))/(-ln(base)))^x)*W(-ln(base))/(-ln(base)) + ln(W(-ln(base))/(-ln(base)))*exp_base[n](1)))

--> Is this formula correct? But...

On Maple -- n = 10 times:
- input
---------------------------------------------------------------------------------------------------------------------------
Digits:=20:
base:=1.35: x:=2.:
log[base](log[base](log[base](log[base](log[base](log[base](log[base](log[base](log[base](log[base]((1 - ln(LambertW(-ln(base))/(-ln(base)))^x)*LambertW(-ln(base))/(-ln(base)) + ln(LambertW(-ln(base))/(-ln(base)))*base^(base^(base^(base^(base^(base^(base^(base^(base^(base^base)))))))))))))))))));
---------------------------------------------------------------------------------------------------------------------------

- output
--------------------------------
5.8512341052940943912
--------------------------------
--> This output is incorrect...

1.35^^2 = 1.4995142162286330979 --> this is correct.


RE: Math.Stackexchange.com question on extending tetration - JmsNxn - 03/31/2021

Hey, nuninho1980.

I'd take everything Anixx posts with a grain of salt. I can vouch for the Newton series he gives, but not the weird Lambert limit. I'm not sure how he's getting that, lol. The Newton series does converge very very slow though, so his method may have just as slow convergence.

Regards, James