Endless fun with power towers

Calculus Level 3

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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