During my testing of the series $\sum\limits_{k=1}^{n} (-1)^k(^kx)$, I found that the sum converges to two limits when $n \to \infty$, for $e^{-e} \lt x \le e^{1/e}$ and oscillates between depending on whether $n$ is even or odd.
Here, $^kx$ is tetration. The notation $^kx$ is the same as $x^{x^{x^{....}}}$, which is the application of exponentiation $k-1$ times. Ex. $^3x=x^{x^x}$.
Questions:
$(1)$ What is the maximum and minimum of $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n} (-1)^k(^kx)$ for even $n$?
$(2)$ What is the maximum and minimum of $\lim\limits_{n\to \infty}\sum\limits_{k=1}^{n} (-1)^k(^kx)$ for odd $n$?
Edit:
Also during my testing in PARI, I observed that the sum seems to converge to two values only in the domain of $e^{-e} \lt x \le e^{1/e}$. I think the reason for this maybe is that, since $^{\infty}x$ converges only for $e^{-e} \lt x \le e^{1/e}$, the sum also converges for the same domain. I would appreciate if someone could explain why the sum converges only for $e^{-e} \lt x \le e^{1/e}$.
from Hot Weekly Questions - Mathematics Stack Exchange
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