# Radius of convergence - power series - misunderstanding

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...?

• Can you provide 'the answer' you have? May 24 '20 at 1:04
• But first of all, $R'$ is the radius of convergent for the given series, according to the Cauchy-Hadamard theorem. May 24 '20 at 1:06
• Its in hebrew and long and thats the start of the proof, is it be of any value to post a long answer in hebrew? And yes i thought that $R'$ is the radius, but where that $R$ came from?
– Alon
May 24 '20 at 1:07
• Ok so maybe its a mistake or used not aas the radius of the serie..
– Alon
May 24 '20 at 1:09
• In both expressions of $R$, replace $a_n$ by $u_n$. $R'$ is correct. May 24 '20 at 1:13

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 $$c_n=a_n^3$$) : $$R'=\frac{1}{\limsup|a_n|^{3/n}}.$$