Tetration (p^^x) for 1

Below is a link to a Desmos application that approximates p^^x (using Knuth up arrow notation) to several digits of accuracy, for p=s^(1/s), e>s>1, x>-2:

https://www.desmos.com/calculator/vlifkv27hp

Is this result new, relevant, and interesting (in the mathematical sense)? If so, what should be my next steps to spread this information through the relevant parts of the mathematical community?

My story: A few years ago I got started trying to describe p^^x for real numbers p and x. I started with 2^^x, which turned out to be really hard, although I have made incremental improvements as I have learned more math. I also saw an article claiming a method to generate points for (sqrt 2)^^x, which although I believed at the time I recently came back to the problem and found it was flawed. In the process I managed to discover a better way to generate the curve of (sqrt 2)^^x, and after that created a more general way to generate reasonable approximations for (s^(1/s))^^x. One caveat is that for the number of terms I have now, the series visually converges well if 1.1<s<1.7. Below 1.1, floating point errors start to become relevant, and above 1.7 the number of terms I have now does not offer good convergence to the actual value. That being said, I believe the way I have it laid out now allows for an easy expansion to higher levels of accuracy and better convergence.

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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