Can you add two radii of convergence?

Question

So I have the function $\frac{1}{x^2-3x+2}$ about $x=-1$ and I am supposed to find the radius of convergence. I did partial fraction decomposition and got $\frac{1}{x^2-3x+2} = \frac{1}{x-2} - \frac{1}{x-1}$. I found the power series for each fraction individually, for $\frac{1}{x-2}$ the power series is $-3^{-1-n}(1+x)^n$ which converges when $|z+1|<3$ and for $- \frac{1}{x-1}$ the power series is $2^{-1-n}(1-x)^n$ which converges when $|x+1| < 2$. Can I say the radius of convergence is 2 since both series converge when $|x+1| < 2$ or am I off base?

Answer 1

It is almost correct, but not entirely. You can do that because you obtained two distinct radii. But if they are both equal to $2$, the only conclusion that you could draw is that the radius of convergence of the sum is at least $2$. But it could be greater than that.

Answer 2

The radius of convergence is the least distance between the point where we are calculating and all the singularities of the function i.e. the minimum between $|-1-1|=2$ and $|-1-2|=3$ which is $2$