Find the series' radius of convergence.
-$$\sum _ { n = 1 } ^ { \infty } \frac { ( x - 3 ) ^ { n } } { ( 4 n ) ! }$$

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 $n+1$ term to the $n$ term. For this series, the ratio of successive terms simplifies to $\frac{|x-3|}{4(n+1)}$, and as $n$ approaches infinity, this ratio approaches 0 for any finite value of $x$. This means the series converges for all $x$, indicating an infinite radius of convergence.

Find the Series' Radius of Convergence ∑n=1∞(x−3)n(4n)!\sum _ { n = 1 } ^ { \infty } \frac { ( x - 3 ) ^ { n } } { ( 4 n ) ! }n=1∑∞​(4n)!(x−3)n​

Find the Series' Radius of Convergence $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 3 ) ^ { n } } { ( 4 n ) ! }$