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.

What is the Maclaurin series of \(f?\)

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

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

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