This is a question that came to my mind as I was reading about power series. Is it possible for two power series to, in some sense, represent the same function, and thus be "identical"? So the definition would go like:
The power series $f(z)=a_0+a_1z+a_2z^2+\ldots$ and $g(z)=b_0+b_1z+b_2z^2+\ldots$ are identical if $f(z)=g(z)$ for all values of $z$ such that $f(z)$ and $g(z)$ converge.
With this definition, if $f(z)$ or $g(z)$ has radius of convergence $0$ and $a_0=b_0$, then they are identical. What if the radius of convergence $R_f$ and $R_g$ are both positive? Does that guarantee that $a_i=b_i$ for all $i\geq 0$?