I did this questions using the Ratio Test which showed that the radius of convergence is the same. I'm not sure if that is correct. (I am having my doubts about c_n becoming c_n+1 for the ratio test
2026-03-29 12:31:31.1774787491
Radius of Convergence Problem solving
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The series $\displaystyle \sum_{n=0}^\infty c_n x^n$ converges if $|x|<R$ and diverges if $|x|>R$ hence the series $\displaystyle \sum_{n=0}^\infty c_n (x^2)^n$ converes if $|x^2|<R\iff |x|<\sqrt R$ and diverges if $|x^2|>R\iff |x|>\sqrt R$ hence its radius is $\sqrt R$.