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?

None of the others 0.75 0.5 0.25 Limit does not exist 1 0.364

1 solution

Otto Bretscher
May 5, 2016

The sequence $x_n$ is recursively given by $x_0=1$ and $x_{n+1}=a^{a^{x_n}}$ for $n\geq 0$ , with the iteration function $f(x)=a^{a^x}$ . We have the fixed point $f(\frac{1}{2})=\frac{1}{2}$ , and a little calculus shows that $0<f'(x)<1$ for $\frac{1}{2}\leq x\leq 1$ . Thus $\frac{1}{2}<x_{n+1}<x_{n}$ for all $n\geq 0$ , and $\lim_{n\to\infty}x_n=\boxed{0.5}$ by continuity of $f(x)$ .

Can you tell me how to find the solution of x= a^a^x without using calculator or hit and trial. I had to use a calculator and I tested out the values to get 1/2 as the answer. Also for odd numbers it would tend to 0.364

Arghyadeep Chatterjee - 3 years ago

I got clearly .364

Pranjit Gautam - 3 years ago

Jesus Christ I meant the limit tends to 0.364 . Whenever you say limit you say tend and not equals . As the variable is tending to a real number the limit should also tend to your answer. I just wanted to know the pen and paper step by step solution to such an exponential eqn as I was unable to solve it like that and I had to use desmos graph to get the value.

Arghyadeep Chatterjee - 3 years ago

@Arghyadeep Chatterjee For an odd number of a’s, the limit would be 0.25.

Tavish Music - 2 weeks, 3 days ago

Endless fun with power towers

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