One more Power Tower

Since we are interested in the even terms only (Herr @Andreas Wendler will take care of the odd terms) , we compute $x_2=a^a= 0.05^{0.05}\approx0.86$ . Now we have the recursive formula $x_{n+2}=a^{a^{x_n}}$ , and we define the iteration function $f(x)=a^{a^x}$ . It turns out that $f(x)$ has three fixed points, where $f(x)=x$ , namely, $c_1\approx 0.14,c_2\approx 0.35, c_3\approx 0.66$ . The function $f(x)$ is increasing and $f(x)<x$ for $x>c_3$ ; see the attached graph. Thus the sequence $x_{2n}$ is decreasing and bounded by $c_3$ below; draw a cob web! We have $\lim_{n\to\infty}x_{2n}=c_3\approx \boxed{0.66}$