Power Tower

$$a\uparrow\uparrow k\equiv \underbrace{{a}^{{a}^{\cdot^{\cdot^{\cdot^{a}}}}}\!\!}_{k}\,,$$

where $\uparrow$ is Knuth's (1976) Arrow Notation.

$$a\uparrow^k n = a\uparrow^{k-1} [a\uparrow^k (n-1)].$$

The infinite power tower $x\uparrow\uparrow\infty=x^{x^{\cdot^{\cdot^\cdot}}}$ converges Iff $e^{-e}\leq x\leq e^{1/e}$ ( $0.0659\leq x\leq 1.4446$).

See also Ackermann Function, Fermat Number, Mills' Constant

References

Knuth, D. E. ``Mathematics and Computer Science: Coping with Finiteness. Advances in our Ability to Compute are Bringing us Substantially Closer to Ultimate Limitations.'' Science 194 1235-1242, 1976.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 11 and 226-229, 1991.