Infinite exponentials We can read a lot of about convergence of series or Infinite products.
E.g. for series.
Following series
$$\sum_{i=1}^\infty a_i$$
is convergent when 
$$\lim_{n\rightarrow\infty}a_n=0$$
and


*

*D'Alembert's criterion


$$\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|<1$$
2. Cauchy's criterion
$$\limsup_{n\rightarrow\infty}\sqrt[n]{\left|a_n\right|}<1$$
3. Raabe–Duhamel's criterion
$$\lim_{n\rightarrow\infty}n\left(\frac{a_{n+1}}{a_n}-1\right)>1$$
For products we can use above criterias for series.
Following product converges
$$\prod_{i=1}^\infty a_i$$
if series
$$\sum_{i=1}^{\infty} \ln{a_i}$$
converges.
What about Infinite exponentials? (I don't know how to write it)
$$a_1^{{a_2}^{{a_3}^{{a_4}^\cdots}}} $$
I know that tetration is particular case of that for 
$$
a_1=a_2=a_3=\cdots=x
$$
Then this infinite exponentials convergence for 
$$
x\in\left<\frac{1}{e^e};\sqrt[e]{e}\right>
$$
Is there general theorem of that? Where can I read about results?
I'm sure that $\lim_{n\rightarrow\infty}a_n=0$ or $\lim_{n\rightarrow\infty}a_n=1$ is not required. Because for $a_i=\sqrt{2}$ we have
$${\sqrt{2}}^{{\sqrt{2}}^{{\sqrt{2}}^{{\sqrt{2}}^\cdots}}}=2 $$

I don't know any theorems for convergence of power towers, but here is food for thought.
Let's define pt(a, n) (pt as short for "power tower") of a sequence a as:


*

*pt(a, 1) = a_1

*pt(a, 2) = a_1^(a_2)

*pt(a, 3) = a_1^(a_2^(a_3))

*...

*pt(a, n) = a_1^(a_2^(a_3^...^(a_n)...))


For lim [n -> oo] pt(a, n) to exist, pt(a, n) should be very near to pt(a, n+1). This requires a_n^a_(n+1) to be very near to a_n, thus a_(n+1) very near 0.
So, a convergence criteria could be a_n nearing 0. Ironically, 0^0 (and 0^0^...^0) are indeterminate forms.
