For a real number x > 1 , let S(x) = _ n=1 ^ n^3 x^n . If S(x) can be expressed in the form x^3 + ax^2 + bx (x - 1)^c , where a, b, and c are integers, then the value of a + b + c is

Step-by-step solution

Given S(x) = _ n=1 ^ n^3 x^n = 1 x + 8 x^2 + 27 x^3 + 64 x^4 + Multiply by 1 x and subtract: S(x) (1 - 1 x ) = 1 x + 7 x^2 + 19 x^3 + 37 x^4 + Multiply by 1 x and subtract again: S(x) (1 - 1 x )^2 = 1 x + 6 x^2 + 12 x^3 + 18 x^4 + Multiply by 1 x and subtract a third time: S(x) (1 - 1 x )^3 = 1 x + 5 x^2 + 6 x^3 + 6 x^4 + Multiply by 1 x and subtract a fourth time: S(x) (1 - 1 x )^4 = 1 x + 4 x^2 + 1 x^3 Simplify the equation: S(x) ( x - 1 x )^4 = x^2 + 4x + 1 x^3 S(x) = x^4 (x - 1)^4 ( x^2 + 4x + 1 x^3 ) S(x) = x(