Find the power series for the function $f ( x ) = e ^ { x ^ { 8 } }$ using the power series for $e ^ { x }$ .
A) $\sum _ { n = 1 } ^ { \infty } \frac { x ^ { 8 ( n + 1 ) } } { ( n + 1 ) ! }$
B) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { 8 n } } { n ! }$
C) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { 8 ( n + 1 ) } } { ( n + 1 ) ! }$
D) $\sum _ { n = 0 } ^ { \infty } \frac { x ^ { 8 n } } { ( n + 1 ) ! }$
E) $\sum _ { n = 1 } ^ { \infty } \frac { x ^ { 8 ( n + 1 ) } } { n ! }$

Question

Quizplus · Accepted Answer

The power series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. Substituting $x^8$ for $x$ gives $\sum_{n=0}^{\infty} \frac{(x^8)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^{8n}}{n!}$, which matches choice B.

Find the Power Series for the Function f(x)=ex8f ( x ) = e ^ { x ^ { 8 } }f(x)=ex8

Find the Power Series for the Function $f ( x ) = e ^ { x ^ { 8 } }$