This is a fascinating topic indeed. And the math behind polynomial approximation is not that deep or unapproachable. You just haven't found an explanation that works for you. Some time ago I tried to make one with a sine as an example: https://wordsandbuttons.online/sympy_makes_math_fun_again.ht...

It doesn't mention Taylor series anywhere but if you only make your system out of N differential equations, you'll get N members of the power series.