$\text { Find the interval of convergence of the power series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 3 n + 1 } } { ( 3 n + 1 ) ! } \text {. (Be sure to }$ include a check for convergence at the endpoints of the interval.)
A) $( - \infty , 4 )$
B) $[ - 4,4 ]$
C) $[ - 3,3 ]$
D) $( 4 , \infty )$
E) $( - \infty , \infty )$

Question

$\text { Find the interval of convergence of the power series } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 3 n + 1 } } { ( 3 n + 1 ) ! } \text {. (Be sure to }$ include a check for convergence at the endpoints of the interval.) 
A) $( - \infty , 4 )$
B) $[ - 4,4 ]$
C) $[ - 3,3 ]$
D) $( 4 , \infty )$
E) $( - \infty , \infty )$

Quizplus · Accepted Answer

The series converges for all real numbers, $x$, because the ratio of successive terms tends to 0 as $n$ approaches infinity, indicating an infinite radius of convergence.

Include a Check for Convergence at the Endpoints of the series