# Endless fun with power towers

Calculus Level 5

For $$a=\dfrac1{16}$$, consider the (finite) power tower,

$\Large x_n=\underbrace{a^{a^{\cdot^{\cdot^{a^a}}}}}_{2n \; a\text{'s}}$

For example, $$x_1=a^a$$ and $$x_2=a^{a^{a^a}}$$.

Find $$\displaystyle \lim_{n\to\infty}x_n$$, to three significant figures.

Bonus What happens if we consider a power tower with an odd number of $$a$$'s?

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