Let $\sum c_n z^n$ be a power series. ($c_n,z_n \in \mathbb{C}$)
Let $\alpha = \lim \sup {|c_n|}^{1/n}$ Then this series is convergent if $|z|<1/{\alpha}$.
Let $\beta = \lim \sup |c_{n+1}/{c_n}|$ Here, series is convergent if $|z|<1/{\beta}$.
Since $\alpha≦\beta$, $|z|<1/{\alpha}$ gives more choice of $|z|$ which makes the series convergent.
It seems like the second one is easier to apply to problems. Is any constraint that makes $\alpha=\beta$?