Find the radius of convergence of the power series. $$\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }$$
A) $\infty$
B) 16
C) 64
D) 0
E) 8

Question

Quizplus · Accepted Answer

The radius of convergence can be found using the ratio test, which involves taking the limit as $n$ approaches infinity of the absolute value of the ratio of the $(n+1)$th term to the $n$th term. For the given series, the $(n+1)$th term is $\frac{(8x)^{2(n+1)}}{(2(n+1))!}$ and the $n$th term is $\frac{(8x)^{2n}}{(2n)!}$. Simplifying the ratio and taking the limit, we find that the series converges for all $x$, meaning the radius of convergence is $\infty$.

Find the Radius of Convergence of the Power Series ∑n=0∞(8x)2n(2n)!\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }n=0∑∞​(2n)!(8x)2n​

Find the Radius of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } \frac { ( 8 x ) ^ { 2 n } } { ( 2 n ) ! }$