Problem 70

Question

Prove that if the power series $\sum_{n=0}^{\infty} c_{n} x^{n}$ has a radius of convergence of $R$, then $\sum_{n=0}^{\infty} c_{n} x^{2 n}$ has a radius of convergence of $\sqrt{R}$.

Step-by-Step Solution

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Answer

The radius of convergence of the series $\sum_{n=0}^{\infty} c_{n} x^{2n}$ is $\sqrt{R}$.

1Step 1: Preliminaries about power series

In general, a power series of the form $\sum_{n=0}^{\infty} c_{n} x^{n}$ converges for all $x$ such that \(|x|

2Step 2: Considering the new power series

Now consider the power series $\sum_{n=0}^{\infty} c_{n} x^{2n}$. If we set $y=x^2$, this series can be rewritten as $\sum_{n=0}^{\infty} c_{n} y^n$, which is the same form as the original series.

3Step 3: Determine the radius of convergence of the new series

Since $y=x^2$, the original interval of convergence \(-R

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