Problem 4 - Entrance Test

Find the Maclaurin series for f(x) = xe^(-x^2).

A. ∑ (-1)^n * x^(2n+1) / n!B. ∑ x^(2n+1) / n!C. ∑ (-1)^n * x^(2n) / n!D. ∑ x^(2n) / n!

Correct: A

We know e^x = ∑ x^n / n!. So, e^(-x^2) = ∑ (-x^2)^n / n! = ∑ (-1)^n x^(2n) / n!. Therefore, xe^(-x^2) = x ∑ (-1)^n x^(2n) / n! = ∑ (-1)^n x^(2n+1) / n!.