Suppose that the radius of convergence of the power series $\Sigma c_{n} x^{n}$ is $R .$ What is the radius of convergence of the power series $\Sigma c_{n} x^{2 n} ?$
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Now, consider the power series $\Sigma c_{n} x^{2n}$. We can rewrite this as $\Sigma c_{n} (x^2)^{n}$. The radius of convergence of this new series is given by $R' = \frac{1}{\limsup_{n\to\infty} \sqrt[n]{|c_n|}}$. However, because we are now considering $x^2$ Show more…
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