The first-order Taylor approximation P1(x) is simply the tangent line at x0,
P1(x) = f(x0)+(x-x0)f '(x0).
The second-order Taylor approximation P2(x) is the parabola that has the same function value and first and second derivatives as f(x) at the point x0. The equation for this approximation is as follows:
T2(x) = f(x0)+(x-x0)f '(x0)+1/2(x-x0)2f ''(x0).
The nth order Taylor approximation is
Σi = 0n[f(i)(x0)/i!]*(x - {x_0})i
Where f(i)(x) indicates the ith derivative of f(x).