# $\limsup$ and $\lim$ and root test.

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 $\lim$?

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So "they" changed from $\limsup$ to $\lim$ once "they" realised that the limit of $2^{-n^2}$ exists.