Which of the following formulas represents the Maclaurin series expansion for (x)?
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Which of the following formulas represents the Maclaurin series expansion for $\cos(x)$?
Show answer & explanation
The Maclaurin series for cos(x) includes only even powers of x with alternating signs. The formula ∑(-1)^n · x^(2n)/(2n)! generates the terms 1 - x²/2! + x⁴/4! - x⁶/6! + ..., which is the correct expansion. Option B is the series for e^x, option C omits the alternating sign factor, and option D is equivalent to e^(-x).
Step-by-step Derivation:
The Maclaurin series is f(x) = ∑(n=0 to ∞) f^(n)(0) · x^n / n!. For cos(x): f(0)=1, f'(0)=0, f''(0)=-1, f'''(0)=0, f''''(0)=1, etc. Derivatives alternate between 0 and ±1, creating the pattern 1, 0, -1, 0, 1, 0, -1, ... This gives only even-powered terms with sign (-1)^n, yielding option A.