Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $$f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$$
A) $5 \left( 1 - 17 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
B) $5 \left( 1 - 9 x ^ { 2 } + \frac { 115 } { 6 } x ^ { 4 } \right)$
C) $5 \left( 1 - 9 x + \frac { 115 } { 6 } x ^ { 4 } \right)$
D) $5 \left( 1 - 9 x ^ { 2 } + \frac { 97 } { 6 } x ^ { 4 } \right)$
E) $5 \left( 1 - 17 x ^ { 2 } + \frac { 67 } { 6 } x ^ { 4 } \right)$

Question

Quizplus · Accepted Answer

The Maclaurin series for $e^{-x^2}$ is $1 - x^2 + \frac{x^4}{2} + \cdots$ and for $\cos 4x$ is $1 - \frac{(4x)^2}{2} + \frac{(4x)^4}{24} + \cdots$. Multiplying these series and simplifying gives the first three nonzero terms as $1 - 9x^2 + \frac{115}{6}x^4$. Multiplying by 5 gives the correct answer.

Use Multiplication or Division of Power Series to Find the First