If f(x)=∑n=0∞cnxnf(x) = \sum_{n=0}^{\infty} c_n x^nf(x)=∑n=0∞cnxn and g(x)=∑n=0∞cn(x−1)ng(x) = \sum_{n=0}^{\infty} c_n (x-1)^ng(x)=∑n=0∞cn(x−1)n, how are their radii of convergence related?
They are the same.
Rg=Rf−1R_g = R_f - 1Rg=Rf−1
Rg=Rf+1R_g = R_f + 1Rg=Rf+1
The relationship depends on cnc_ncn.