> Infinite series can't really be calculated to completion using a computer,

The sum of an infinite divergent series cannot be calculated with or without a computer.

The sum of an infinite convergent series can be calculated with:

  1/(a-r)

Sequence > Limits and convergence: https://en.wikipedia.org/wiki/Sequence#Limits_and_convergenc...

Limit of a sequence: https://en.wikipedia.org/wiki/Limit_of_a_sequence

SymPy docs > Limits of Sequences: https://docs.sympy.org/latest/modules/series/limitseq.html

> Provides methods to compute limit of terms having sequences at infinity.

Madhava-Leibniz formula for π: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

And how thousands of lines of C, below, to actually get the work done?