Maclaurin Series

Calculus Level 2

A function $f$ is defined for all real numbers and satisfies

$f(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -1,\quad f^{(4)}(0) = 0,$

and, in general, $f^{(k)}(0) = (-1)^{k}$ if $k$ is divisible by 3, and $f^{(k)}(0) = 0$ otherwise.

Clarification: In the answer choices, $!$ denotes the factorial function. For example, $8! = 1\times2\times3\times\cdots\times8$ .

\sum_{k=1}^\infty \frac{(-1)^k}{k!} x^{3k}

\sum_{k=0}^\infty \frac{(-1)^k}{(3k)!} x^{3k}

\sum_{k=1}^\infty (-1)^{3k} x^{k}

\sum_{k=0}^\infty (-1)^{3k} x^{3k}

\sum_{k=0}^\infty \frac{(-1)^k}{k!} x^{3k}