Suppose we're asked to find the radius of convergence of a power/Taylor series for some function. Is there a way to know or guess the radius of convergence without computing the $f^{(n)}(0)$'s (or $f^{(n)}(a)$'s)? The one I encountered was some rational function that was something like $$f(x) = \frac{1+x^3}{x-2}$$
I used the ratio test to get
$$\lim_{n \to \infty} |\frac{f^{(n+1)}(0)}{f^{(n)}(0)}||\frac{n!}{(n+1)!}||\frac{x^{n+1}}{x^n}|$$
$$= |\frac{f^{(n+1)}(0)}{f^{(n)}(0)}| \frac{1}{n+1} |x|$$
I guessed the limit would be zero for any $x$ and hence the radius is infinite. Since $f(x)$ isn't defined at $x=2$, I'm guessing I was wrong. Even if I remembered the function wrong, I believe the denominator wasn't strictly nonzero.
I'm not sure the shortest way to go about this is to compute the first few $f^{(n)}(0)$'s and then guess a pattern. Is there perhaps some kind of theorem for radii of convergence, $f^{(n)}(0)$'s or $\frac{f^{(n+1)}(0)}{f^{(n)}(0)}$'s for certain functions (e.g. rational, continuous on its domain, etc)?
P.S. You may use complex analysis, but I haven't taken that yet.