Whats the difference between using $\limsup$ and $\lim$ when using the root test to find the radius of convergence? For finding the radius of convergence of $\displaystyle\sum\limits_{n=0}^\infty2^{-n^2}z^n$, it was written:
$\begin{align} \limsup_{n\rightarrow\infty}|c_n|^\frac{1}{n}&\\ &=\lim_{n\rightarrow\infty}|2^{-n^2}|^\frac{1}{n}\\ &=\lim_{n\rightarrow\infty}|2^{-n}|=0\\ &&\text{So radius of convergence is }\infty. \end{align}$
How did (can) they change from $\displaystyle\limsup_{n\rightarrow\infty}|c_n|^\frac{1}{n}$ to $\displaystyle\lim_{n\rightarrow\infty}|2^{-n^2}|^\frac{1}{n}$? Whats the difference between the limits of $\limsup$ and $\lim$?