Find the Taylor series generated by f at x = a.
-$$f ( x ) = e ^ { x } , a = 10$$
A) $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { \mathrm { e } ^ { 10 } ( \mathrm { x } - 10 ) ^ { \mathrm { n } } } { \mathrm { n } ! }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { e ^ { 10 } ( x - 10 ) ^ { n } } { ( n + 1 ) ! }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { \mathrm { e } ^ { 10 } ( \mathrm { x } - 10 ) ^ { \mathrm { n } + 1 } } { \mathrm { n } ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { e ^ { 10 } ( x - 10 ) ^ { n + 1 } } { ( n + 1 ) ! }$

Question

Quizplus · Accepted Answer

The Taylor series of $f(x) = e^x$ at $x = a$ is given by $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$. Since the $n$th derivative of $e^x$ is $e^x$ for all $n$, and evaluating at $a = 10$ gives $e^{10}$, the correct series is $\sum_{n=0}^{\infty} \frac{e^{10}(x-10)^n}{n!}$.

Find the Taylor Series Generated by F at X = f(x)=ex,a=10f ( x ) = e ^ { x } , a = 10f(x)=ex,a=10

Find the Taylor Series Generated by F at X = $f ( x ) = e ^ { x } , a = 10$