Math.Stackexchange.com question on extending tetration

[Daniel]

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

[JmsNxn]

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

[nuninho1980]

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.

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.

[JmsNxn]

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