Answer to Question #4194 in Calculus for John

Question

Prove that integral(bottom=a, top=b) (x[sup]2[/sup]+1)e[sup]-x^2[/sup]& dx & >= & (e[sup]-a^2& [/sup]- e[sup]-b^2[/sup])

Answer 1

Denote x_1 = 1/2
x_2 = -1/4
x_3 = 1/8
x_4 = -1/16
Then
x₁ = 1/2 = 1/(2¹) = (-1)¹⁺¹/(2¹)x₂ = -1/4 = -1/(2²) = (-1)²⁺¹/(2²)x₃ = 1/8 = 1/(2³) = (-1)³⁺¹/(2³)x₄ = 1/16 = 1/(2⁴) = (-1)⁴⁺¹/(2⁴)Thus the general formula for x_n is the following
x_n = -1ⁿ⁺¹/ 2ⁿ