Difficulty deriving Maclaurin series for (1+x)^a where a is fractional

I have been working on the power series expansion of (1+x)^a with a fractional exponent, say a = 1/3. I know that the generalized binomial theorem gives (1+x)^a = Σ_{k=0}^{∞} binomial(a, k) x^k, where binomial(a, k) = a(a-1)...(a-k+1)/k!. However, when I try to compute the coefficients explicitly, I keep running into confusion about signs and convergence radius. For instance, what is the second coefficient when a = 1/3? It seems to come out negative, but my calculator gives a positive number for small x, so I'm not sure if I've set up the product correctly. Also, I want to confirm that the radius of convergence remains 1 for non-integer a – is there a concise proof you could point me toward? Finally, would an alternative approach using Taylor's formula with the nth derivative at zero be more reliable for higher terms? Any step-by-step guidance would be greatly appreciated, and I apologize in advance for the length of my question – I just want to make sure I understand every nuance. Thanks in advance for your patience!