Radius of convergence - power series - misunderstanding

Question

What is the radius of convergence of:

$$ \sum_{n = 0}^{\infty}a_n^3z^n $$

I know that the formal calculation of the radius is by Cauchy-Hadamard:

$$ R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{a_n}} $$

So I don't understand why the answers show $2$ radiuses:

$$ R = \frac{1}{\limsup\sqrt[n]{|a_n|}} $$

and:

$$ R' = \frac{1}{\limsup \sqrt[n]{|a_n|^3}} $$

Why are there $2$ radiuses? What is this $R$, it's not exactly as the formula is...?

Answer 1

I don't know the literature, but I guess the first $R$ is from Cauchy-Hadamard theorem :

Consider $f(z)=\sum_{n=0}^\infty c_nz^n$. Then the radius of convergence for $f$ is $R=1/\limsup_{n\rightarrow\infty}|c_n|^{1/n}$.

Then the author of your 'the answer' did not want to use same notation for the radius of convergence of the given problem, so used $R'$ notation. Anyway, if you have $f(x)=\sum_{n=0}^\infty a_n^3z^n$, then it's radius of convergence is (note that $c_n=a_n^3$) : $$R'=\frac{1}{\limsup|a_n|^{3/n}}.$$