I would like to further develop the result of this question, that I summarize as follows:
Theorem. If $f=\sum a_k x^k$ and $g=\sum b_k x^k$ are power series with radius of convergence $R_f$ and $R_g$, then $f+g$ has radius of convergence at least $\min(R_f,R_g)$ with equality if $R_f\ne R_g$.
My conjecture is that
Theorem (conjecture). If (maintaining all the assumptions of the claim above) $R_f = R_g$, $f+g\ne 0$ and $a_k\ne b_k$ for infinite many $k$, then $R_{f+g}=\min(R_f,R_g)$ also if $R_f=R_g$.
Another shorter formulation could be: if $f-g$ is not a polynomial, then $R_{f+g}=\min(R_f,R_g)$.
Q: Can someone confirm my conjecture?
Consider $\sum x^{n}$ and $\sum (1+\frac 1{n!}) x^{n}$. This is a counter example. $f-g$ is entire in this case.
I fact radius convergence of $f-g$ can be any number greater than the minimum of $R_f$ and $R_g$.