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 $a_n=c_{n/2}$ if $n$ is even, and $a_n=0$ if $n$ is odd. Then we have
$$\sum_{n=0}^\infty a_n(x^2)^n=\sum_{n=0}^\infty c_{2n}x^{2n}.$$
By the ratio test, the radius of convergence of the series on the left-hand side of the above equation is
$$\lim_{n	o\infty}\left|\frac{a_{n+1}x^2}{a_nx^2}ight|=\lim_{n	o\infty}|c_{n+1}/c_n|^{1/2}=\frac{1}{9},$$
so the radius of convergence of $\sum_{n=0}^\infty c_{2n}x^{2n}$ is $\boxed{3}$.

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 $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$