This one weird thing that bugs me about summation and the like

Maybe https://en.wikipedia.org/wiki/Iterated_function#Some_formulas_for_fractional_iteration and https://en.wikipedia.org/wiki/Ramanujan_summation can bring some light, but i think it's easy find oneself lost in greater levels of abstraction, without good references or specific examples.

The abstraction you hint at is a standard example in Scheme or LISP programming. This is from page 64 of Abelson and Sussman's classic Structure and Interpretation of Computer Programs (http://web.mit.edu/alexmv/6.037/sicp.pdf):

Exercise 1.32.

Show that sum and product (exercise 1.31) are both special cases of a still more general notion called accumulate that combines a collection of terms, using some general accumulation function:

(accumulate combiner null-value term a next b)

Accumulate takes as arguments the same term and range specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a null-value that specifies what base value to use when the terms run out. Write accumulate and show how sum and product can both be defined as simple calls to accumulate.

This construction doesn't address any of your questions about convergence or integration. It can be further generalized to deal with potentially infinite streams of values to be combined, should you wish to pursue that.