# Find a sequence such that this tower of of exponent is convergent

Context

• We already know that if we take a sequence $(x_n)\in{\mathbb R_+^*}^{\mathbb N}$ such that

$$x_n=O\left(\frac 1{n^2}\right)$$

then

$$\sum_{n=0}^\infty x_n <+\infty.$$

• We also now that if we take for instance for all $n\in \mathbb N^*$

$$x_n=1+\frac 1{n^2}$$

then

$$\prod_{n=0}^\infty x_n <+\infty.$$

Can we find a sequence $(x_n)\in(1,\infty)^{\mathbb N}$ such that

$${{x_0}^{{x_1}^{{x_2}^{x_3}}}}^{\dots} = {{x_0}^{\left({x_1}^{\left({x_2}^{x_3^\cdots}\right)}\right)}}$$

is convergent?

I think the answer is yes if we take $(x_n)$ such that $\lim_{n\to\infty} x_n=1$ and $(x_n)$ converges to $1$ really fast. But I don't know how to exhibit such a sequence.

• With $(x_n)\in(1,\infty)^{\mathbb N}$ do you mean that it must take integer values greater then one? – Jan Dec 22 '16 at 18:53
• @Jan Yes, that is what I meant. – E. Joseph Dec 22 '16 at 20:57

The lame answer is to have $x_0=1$ or $x_0=0,x_1>0$. Then the result is trivial.
If $x_n=x_0$ for all $n$ and $x_0>0$, then it converges iff $e^{-e}\le x_0\le e^{e^{-1}}$, which actually does not require $\lim_{n\to\infty}x_n=1$. These are found in the Wikipedia for tetration. More information on the exact nature of convergence for $x_n\in\mathbb C$ may be found here and here and here.
From here, you can show that for any sequence with $e^{-e}\le x_0\le e^{e^{-1}}$ that is monotonically approaching $1$ will converge.