Find the Maclaurin series for $f ( x )$ using the definition of the Maclaurin series.
$$f ( x ) = x \cos ( 4 x )$$
A) $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
B)
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$
C)
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$
D)
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$
E)
$\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 4 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$

Question

Quizplus · Accepted Answer

Using the definition of the Maclaurin series, we have

\begin{align*}
f(x) &= x \cos (4x) \
&= x \sum_{n=0}^\infty \frac{(-1)^n (4x)^{2n}}{(2n)!} \
&= \sum_{n=0}^\infty \frac{(-1)^n 4^{2n} x^{2n+1}}{(2n)!}
\end{align*}

Thus, the correct choice is (D).

Find the Maclaurin Series For f(x)f ( x )f(x) Using the Definition of the Maclaurin Series

Find the Maclaurin Series For $f ( x )$ Using the Definition of the Maclaurin Series