If $$\sum_{n=1}^{\infty}a_nx^n$$ has radius of convergence R (with 0

Question

If $$\sum_{n=1}^{\infty}a_nx^n$$ has radius of convergence R (with 0<R<infinity), what is the radius of convergence of $$\sum_{n=1}^{\infty}a_nx ^{5n}$$? 
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My steps: 
$$\lim_{n \rightarrow \infty}|\frac{a_{n+1}x^{n+1}}{a_nx^n}|=\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_n}|x|$$ which converges for |x| <1 so the limit of ratio (a_n+1)/a_n = R?

So, $$\lim_{n \rightarrow \infty}|\frac{a_{n+1}x^{5(n+1)}}{a_nx^{5n}}|=\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_n}|x^5|$$ so that |x| = R^(1/5), so the radius is $$\sqrt[5]{R}$$?

kirbykirby · Answer

actually, |x| < R^(1/5) for the last line I meant

anonymous · Answer

i think this is right.