The Maclaurin series representation of $f ( x ) = e ^ { \frac { x } { 3 } }$ is

A) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 ^ { n + 1 } }$
B) $\sum _ { n = 0 } ^ { \infty } ( - 3 ) ^ { n } x ^ { n }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { 3 ^ { n } x ^ { n } } { n ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 ^ { n } n ! }$
E) $- \sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n } x ^ { n } } { n }$

Question

The Maclaurin series representation of $f ( x ) = e ^ { \frac { x } { 3 } }$ is

A) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 ^ { n + 1 } }$
B) $\sum _ { n = 0 } ^ { \infty } ( - 3 ) ^ { n } x ^ { n }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { 3 ^ { n } x ^ { n } } { n ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 3 ^ { n } n ! }$
E) $- \sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n } x ^ { n } } { n }$

Quizplus · Accepted Answer

The Maclaurin series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. For $e^{\frac{x}{3}}$, replace $x$ with $\frac{x}{3}$ to get $\sum_{n=0}^{\infty} \frac{(\frac{x}{3})^n}{n!} = \sum_{n=0}^{\infty} \frac{x^n}{3^n n!}$.

The Maclaurin Series Representation Of f(x)=ex3f ( x ) = e ^ { \frac { x } { 3 } }f(x)=e3x​

The Maclaurin Series Representation Of $f ( x ) = e ^ { \frac { x } { 3 } }$