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
$${\sqrt{2}}^{{\sqrt{2}}^{{\sqrt{2}}^{{\sqrt{2}}^\cdots}}}=2 $$