Let $f$ be a degree $d$ polynomial, and $N(x)=x-f(x)/f'(x)$ be its Newton root-finding iteration. Suppose $f(c)=0$ and consider the Taylor series of $M_c(z)=N(c+z)-c$: what is its radius of convergence?
For example, $$f(x)=x^2+x$$ $$r(M_{-1})=\frac{1}{2}$$ and for $$f(x)=x^3+2x^2+x+1$$ I tried SymPy thus:
from sympy import limit, roots, simplify, oo
from sympy.abc import c, m, n, z
from sympy.series.formal import rational_algorithm
cs = roots(c ** 3 + 2 * c ** 2 + c + 1, c)
M = (2 * z ** 3 + (3 * c + 2) * z ** 2) / (3 * z ** 2 + (6 * c + 4) * z + 3 * c ** 2 + 4 * c + 1)
a, independent_term, order = rational_algorithm(M, z, n, full=True)
print(f"a_n = {a}")
r = abs(simplify((a.subs(n, m) / a.subs(n, m + 1))))
print(f"r_m = {r}")
for t in cs:
print(f"R({t.evalf(5)}) = {limit(r.subs(c, t), m, oo).evalf(5)}")
Which gives $r(M_{-1.7549\ldots}) = 0.75488\ldots$ for the real root but fails with the conjugate root pair (see https://stackoverflow.com/questions/67023298/result-in-sympy-depends-on-sign-of-a-complex-number). Numerically it seems $r(M_{-0.12256\ldots \pm i 0.74486\ldots}) = 0.77411\ldots$ but that doesn't make sense to me as it's bigger than the imaginary component of the root, and there is a fractal boundary in between.
Context: investigating Newton's method applied to the Mandelbrot set, ultimately trying to figure out if the immediate basin is larger than the atom domain (or a constant factor thereof). I conjectured that $r(M_c)$ is related to the size of the immediate basin, but it seems the relationship isn't straightforward, if it exists at all...