Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$.
A) 252
B) 3
C) 1
D) 256
E) 16

Question

Quizplus · Accepted Answer

Let $R$ be the radius of convergence of the power series $\sum_{n=0}^{\infty}c_nx^n$. By the ratio test: $$\lim_{n o\infty}\frac{|c_{n+1}x^{n+1}|}{|c_nx^n|}=\lim_{n o\infty}|x|\cdot \frac{|c_{n+1}|}{|c_n|}=\frac{|x|}{R}$$ This limit exists and is finite if $|x|R$. Thus, the original series converges absolutely if $|x|R$. Now consider the power series $\sum_{n=0}^{\infty}c_nx^{2n}$. By comparison to the series $\sum_{n=0}^{\infty}|c_nx^n|$ (which has the same radius of convergence as $\sum_{n=0}^{\infty}c_nx^n$), we see that $\sum_{n=0}^{\infty}c_nx^{2n}$ converges absolutely if $|x^2|\sqrt{R}$, then $|x^2|>R$, and the series $\sum_{n=0}^{\infty}c_nx^{2n}$ diverges by the same argument as before. Therefore, the radius of convergence for $\sum_{n=0}^{\infty}c_nx^{2n}$ is $\sqrt{R}$. Substituting $R=9$, we get that the radius of convergence for $\sum_{n=0}^{\infty}c_nx^{2n}$ is $\sqrt{9}=3$, so the answer is $\boxed{ extbf{(B)}}$.

Suppose That the Radius of Convergence of the Power Series ∑n=0∞cnxn\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }∑n=0∞​cn​xn

Suppose That the Radius of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$