I have a somewhat theoretical question to the definition of the radius radius of convergence of infinite power series. According to the definition for a power series $\sum_{n=0}^\infty a_nx^n$ radius of convergence $\rho \ge0$. For $|x|<\rho$ the series converges at $-\rho<x<\rho \\$.
Do I understand it correctly, that if $\rho=0$, the series won't converge?
I browsed several articles on Wikipedia, but didn't find an answer.
No, you don't.
The sum always converges absolutely for $-\rho < x < \rho$, and may or may not converge when $|x|=\rho$. Clearly if $\rho=0$, the series converges when $x=0$ (and only then).