The nature of infinity and irrational numbers (among other things) always gave me trouble. But recently I learned to think about infinite sequences in terms of recursive algorithms and their time evolution - and it turned out to be much easier to accept.
I know that in principle, general Cauchy sequences can't be defined recursively (since almost all real numbers are uncomputable), but I'm fine with restricting myself to computable numbers only.
So I wanted to ask for a reference on this topic - where the general properties of recursive algorithms are connected to the properties of infinite sequences of various nature.
Update:
I found some good reference on the topic of generalization of infinite series, products, fractions etc. Dixon Jones published two papers where he introduced the term 'continued powers', which have the general form:
$$ \left(a_1+b_1\left(a_2+b_2 \left(a_3+b_3... \right)^{p_3} \right)^{p_2} \right)^{p_1} $$
It is easy to obtain from this expression infinite series, infinite products, continued fraction etc. Or any mix of the above.
However, there is no general convergence theorems as far as I can see. Only a particular case was investigated:
$$ \left(a_1+\left(a_2+\left(a_3+... \right)^{p} \right)^{p} \right)^{p} $$
His papers can be found here: first and second.
And moreso, even the most general expression for 'continued powers' does not include any possible recursive algorithm.
There is a Kolmogorov book on algorithms and computational complexity for example, does it deal with this topic?