ln(1+z) = Σn=1,∞ (-1)n-1 zn/n so
g(x) = ∑n=1,∞ (-1)n-1 x2(n-1)/n
To get an idea about the radius of curvature, consider what happens for purely imaginary values of x,that is let x = i b for real b.
g(i b) = ∑n=1,∞ b2(n-1)/n for b = 1 this series diverges logarithmically. for b >1 the divergence is even faster.
So the radius of curvature is |x| <1