# 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, $$8! = 1\times2\times3\times\cdots\times8$$.

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