In the Maclaurin series for e^x, what is the coefficient of x^2?

• A. 0
• B. 1/2
• C. 1
• D. 2

Answer: (B)

Maclaurin series express e^x as a power series around 0, with each term’s coefficient given by f^(n)(0)/n! for f(x) = e^x. Since every derivative of e^x is e^x, evaluating at 0 gives f^(n)(0) = 1 for all n. Therefore the coefficient of x^2 is f''(0)/2! = 1/2. Expanding explicitly shows e^x = 1 + x + x^2/2 + x^3/6 + …, so the x^2 term carries the coefficient 1/2.