Find the radius of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \frac { ( 5 x ) ^ { 2 n } } { ( 2 n ) ! }$
A)0
B)5
C)10
D)25
E) $\infty$

Question

Quizplus · Accepted Answer

The radius of convergence of the given power series can be found using the ratio test. The general term of the series is $a_n = \frac{(5x)^{2n}}{(2n)!}$. Applying the ratio test involves taking the limit as $n$ approaches infinity of the absolute value of $a_{n+1}/a_n$. This simplifies to the limit of $\frac{(5x)^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{(5x)^{2n}}$, which further simplifies to $\frac{(5x)^2}{(2n+1)(2n+2)}$. As $n$ approaches infinity, this ratio approaches 0 for any finite value of $x$, indicating that the series converges for all $x$. Therefore, the radius of convergence is $\infty$.

Find the Radius of Convergence of the Power Series ∑n=0∞(5x)2n(2n)!\sum _ { n = 0 } ^ { \infty } \frac { ( 5 x ) ^ { 2 n } } { ( 2 n ) ! }∑n=0∞​(2n)!(5x)2n​

Find the Radius of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } \frac { ( 5 x ) ^ { 2 n } } { ( 2 n ) ! }$