The interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }$ is

A)(-4,-2)
B)[-4,-2)
C)(-4,-2]
D)[-4,-2]
E)(-$\infty$, $\infty$ )

Question

The interval of convergence of the power series $\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }$ is

A)(-4,-2)
B)[-4,-2)
C)(-4,-2]
D)[-4,-2]
E)(-$\infty$, $\infty$ )

Quizplus · Accepted Answer

The given series is a Taylor series expansion of $e^{x+3}$ centered at $x=-3$, which converges for all $x$, hence the interval of convergence is $(-\infty, \infty)$.

The Interval of Convergence of the Power Series ∑n=0∞(x+3)nn!\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }∑n=0∞​n!(x+3)n​

The Interval of Convergence of the Power Series $\sum _ { n = 0 } ^ { \infty } \frac { ( x + 3 ) ^ { n } } { n ! }$