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Find the coefficient of x⁴ in the power series expansion of (e^x - 1)².

1. 3/8
2. 5/12
3. 7/12
4. 5/24

Correct answer: 7/12

Solution

Expanding (e^x - 1)² = e^(2x) - 2e^x + 1. Coefficients of x⁴: from e^(2x) -> (2x)⁴/4! = 16/24 = 2/3; from -2e^x -> -2/4! = -2/24 = -1/12; from 1 -> 0. Total = 2/3 - 1/12 = 8/12 - 1/12 = 7/12.

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