Evaluate: _ x 0 [3] 1+x - [3] 1-x x

Step-by-step solution

Let f(x) = (1+x)^ 1/3 - (1-x)^ 1/3 x . Using the binomial expansion (1+u)^n = 1 + nu + n(n-1) 2! u^2 + for small u , we have: (1+x)^ 1/3 = 1 + 1 3 x + 1 3 ( 1 3 -1) 2 x^2 + = 1 + 1 3 x - 1 9 x^2 + (1-x)^ 1/3 = 1 - 1 3 x + 1 3 ( 1 3 -1) 2 (-x)^2 + = 1 - 1 3 x - 1 9 x^2 + Substituting these into the expression: _ x 0 (1 + 1 3 x - 1 9 x^2 + ) - (1 - 1 3 x - 1 9 x^2 + ) x _ x 0 2 3 x + O(x^3) x _ x 0 ( 2 3 + O(x^2)) = 2 3 Answer: 2 3