Apply Taylor's Theorem to find the power series centered at $c = 12$ for the function $f ( x ) = e ^ { x }$ .
A) $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n } } { n ! }$
B) $\sum _ { n = 1 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { ( n + 1 ) ! }$
C) $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { n } } { n ! }$
D) $e ^ { 12 } \sum _ { n = 0 } ^ { \infty } \frac { ( x ) ^ { n + 1 } } { ( n + 1 ) ! }$
E) $\sum _ { n = 0 } ^ { \infty } \frac { ( x - 12 ) ^ { 2 n } } { 2 n ! }$

Question

Quizplus · Accepted Answer

Taylor's Theorem allows us to express a function as an infinite sum of its derivatives at a point. For $f(x) = e^x$, all derivatives are $e^x$, and evaluating at $c = 12$ gives $e^{12}$ for each term. The correct series is $e^{12} \sum_{n=0}^{\infty} \frac{(x-12)^n}{n!}$, which matches choice C.

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