Consider the series $\displaystyle \sum _{n=0}^{\infty} (-1)^n z^{2n+1}=z-z^3+z^5-z^7+...$

We notice that $\displaystyle a_k=0$ if $\displaystyle k=2n $ and $\displaystyle a_k=(-1)^n$ if $\displaystyle k=2n+1$, thus $\displaystyle \nexists \lim |a_n|^{\frac{1}{n}}=L$.

However the series has a radius of convergence.

But if I'm not wrong, the radius of convergence is defined as $\displaystyle R=\frac{1}{\lim |a_n|^{\frac{1}{n}}}$. So how can $\displaystyle R$ exist when $\displaystyle L$ does not exist?