Infinite towers of some familiar constants

Calculus Level 4

Among the following infinite tetrations of numbers given, how many of them converge to a particular value?

(A) \(\large e^{e^{e^{\cdot^{\cdot^{\cdot}}}}}\)

(B) \({\sqrt5}^{{\sqrt5}^{{\sqrt5}^{\cdot^{\cdot^{\cdot}}}}}\)

(C) \({\sqrt2}^{{\sqrt2}^{{\sqrt2}^{\cdot^{\cdot^{\cdot}}}}}\)

(D) \({\sqrt3}^{{\sqrt3}^{{\sqrt3}^{\cdot^{\cdot^{\cdot}}}}}\)

(E) \(\large \varphi^{\varphi^{\varphi^{\cdot^{\cdot^{\cdot}}}}}\)

(F) \(\large \pi^{\pi^{\pi^{\cdot^{\cdot^{\cdot}}}}}\)

(G) \(\large {\sqrt e}^{{\sqrt e}^{{\sqrt e}^{\cdot^{\cdot^{\cdot}}}}}\)

(H) \(\large {\sqrt \pi}^{{\sqrt \pi}^{{\sqrt \pi}^{\cdot^{\cdot^{\cdot}}}}}\)

(I) \(\large {\sqrt \varphi}^{{\sqrt \varphi}^{{\sqrt \varphi}^{\cdot^{\cdot^{\cdot}}}}}\)

(J) \(\large \gamma^{\gamma^{\gamma^{\cdot^{\cdot^{\cdot}}}}}\)

(K) \(\large {\sqrt \gamma}^{{\sqrt \gamma}^{{\sqrt \gamma}^{\cdot^{\cdot^{\cdot}}}}}\)

About some of the above mentioned constants:

  • \(e \approx 2.71828...\), denotes the Euler's number.
  • \(\pi \approx 3.14159...\), called as 'pi' denotes the ratio of the circumference of a circle to its diameter.
  • \(\varphi = \dfrac{1+\sqrt5}{2} \approx 1.61803...\), denotes the Golden Ratio.
  • \(\gamma \approx 0.57721...\), denotes the Euler Mascheroni constant.

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