Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be any function that satisfies $$\lim\limits_{n\to\infty} \left(e^{\frac1e} + \frac1n\right)^{\wedge\wedge}\left[(10 n)^{1/2} + n^{A(n)} + C+o(1)\right] – n = 0$$ where $C $ is a constant. Then $\lim\limits_{n\to\infty} A(n) = \frac1e $ Conjectured by […]

I recently posted this, post containing a series of questions concerning the integration of ${x^{x^{x^x}}}$. In order to do so, I wrote ${x^{x^{x^x}}}$ as the following infinite summation: $$1 + \sum_{n=0}^{\infty}\sum_{k=0}^{n}\bigg(\frac{x^{n+1} \log^{n+1+k}(x)C_{kn}}{\Gamma(n+2)}\bigg)$$ I got to this summation using a “generalized Puiseux series”. I mention in the post that I am unsure of how to calculate […]

I was studying tetrations, or “power towers”, and I found a decently well-known fact. The last $k-1$ digits of $^k 3 = 3^{3^{\vdots^{3}}} (k \text{ threes)}$ remain constant, for all numbers $^a 3$ with $a \ge k$ (see here for more). Why is this true? The link shows an ad-hoc proof for the last two […]

This question is about where I made my mistake in the computation of a limit. It relates to An answer I gave that confused me. The question to which I gave the (partial) answer is related to tetration but my mistake is probably a simple general one ( considering tetration as complicated ). Here is […]

For a long time (a couple of years) I’m following the Q&A’s about “half-iterate of $\exp(x)$” etc. where there exists a $\mathbb C \to \mathbb C$ due to Schröder’s method, but also a $\mathbb R \to \mathbb R$ for fractional heights $h$ due to Hellmuth Kneser. I would like to understand the latter method of […]

To start off, I was looking at the following ingeniously made form of the Gamma function: $$\Gamma(z+1)=\lim_{n\to\infty}\frac{n!(n+1)^z}{(1+z)(2+z)\cdots(n+z)}$$ which lies on the back of $$1=\lim_{n\to\infty}\frac{n!(n+1)^z}{(n+z)!}$$ for all integer $z$. One them multiplies through by $z!$ and use the recursive formula for the factorial to reach the above formula. In the same light, I was wondering if […]

This question already has an answer here: Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers? 2 answers How to evaluate fractional tetrations? 4 answers

$$n=a^{(a+1)^{(a+2)^{(a+3)\cdots}}}$$ How would one go about solving in this equation? I am more used to solving equations in this form: $$n=a^{a^{a^{a\cdots}}}$$ Which you solve in this form: $$a^n=n$$ But how would you solve that equation at the top of the page though? If you were curious, and I know that SE likes what I have […]

Let $f(0)=1$ and $f(x+1)=2^{f(x)}$ Also let f be infinitely differentiable. Then does f exist and is it unique? If f is merely continuous, then any continuous function such that f(0)=1 f(1)=2 satisfies the conditions(if f is defined in [0,1] ,we can use the property to define it everywhere else). Similar things can be said for […]

We know how to add with decimals. We know how to multiply with decimals. We know how to exponentiate with decimals. Do we know how to work with decimals for power towers? for example, can we deal with the following expression: 3↑↑3.5? If so, how would you calculate it and can this be done in […]

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