Maclaurin Series

Calculus Level 2

A function ff is defined for all real numbers and satisfies

f(0)=1,f(1)(0)=0,f(2)(0)=0,f(3)(0)=1,f(4)(0)=0,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)kf^{(k)}(0) = (-1)^{k} if kk is divisible by 3, and f(k)(0)=0f^{(k)}(0) = 0 otherwise.

What is the Maclaurin series of f?f?

Clarification: In the answer choices, !! denotes the factorial function. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

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