# 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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