Find the radius of convergence and the interval of convergence of the power series. $$\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$$
A) $R = 7 , I = ( - 7,7 )$
B) $R = 0 , I = \{ 0 \}$
C) $R = 7 , I = [ - 7,7 ]$
D) $R = \infty , I = ( - \infty , \infty )$

Question

Quizplus · Accepted Answer

The radius of convergence is found using the ratio test, which in this case leads to an infinite radius due to the $n!$ in the denominator growing faster than any power of $n$. Thus, the series converges for all $x$, making the interval of convergence $(- \infty , \infty )$.

Find the Radius of Convergence and the Interval of Convergence ∑n=0∞(7x)nn!\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }n=0∑∞​n!(7x)n​

Find the Radius of Convergence and the Interval of Convergence $\sum _ { n = 0 } ^ { \infty } \frac { ( 7 x ) ^ { n } } { n ! }$