Apply Taylor's Theorem to find the power series centered at $c = 0$ for the function $f ( x ) = e ^ { 12 x }$ .
A) $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { ( n + 1 ) ! }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { ( n + 1 ) ! }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n } } { n ! }$
D) $\sum _ { n = 1 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( 12 x ) ^ { n + 1 } } { n ! }$

Question

Quizplus · Accepted Answer

The power series expansion of $e^{ax}$ is given by $\sum_{n=0}^{\infty} \frac{(ax)^n}{n!}$. For $f(x) = e^{12x}$, the series becomes $\sum_{n=0}^{\infty} \frac{(12x)^n}{n!}$, matching choice C.

Apply Taylor's Theorem to Find the Power Series Centered At c