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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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For given sequence, the limit may or may not exist. The limsup always exists. Whenever the limit exists it is equal to the limsup and also to the liminf.

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

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  • $\begingroup$ okay, thanks for your answer! $\endgroup$ – Derrick May 24 '12 at 13:32

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