My professor defined radius of convergence of power series $\sum_{n=0}^{\infty}c_nx^n$ (where $c_n\in\Bbb R$) as $R=\sup\{r\ge0:(c_nr^n )\text{ is bounded}\}$. I have not seen this kind of definition before, can someone motivate. In particular, it seems very different from ordinary definition of radius of convergence ${\displaystyle \sup \left\{|x|\ \left|\ \sum _{n=0}^{\infty }c_{n}x^{n}\ {\text{ converges }}\right.\right\}}$
He has shown that $\frac{1}{R}=\limsup_{n\to\infty}|c_n|^{\frac1n}$, but this does not clear the idea. In particular, I can't understand why boundedness of $c_nr^n$ has anything to do with convergence, I would rather think about $\sum_{k=0}^n c_nx^n$'s boundedness.