MACLAURIN SERIES : Power series for Sin x

Let     f(x) =  sin x   so    f(0)    =  0
        f '(x)  = cos x   so    f '(0)  =  1
        f ''(x) = - sin x  so    f ''(0) =  0
        f '''(x) = -cos x  so    f '''(0)= -1
        .
        .
        .

using Maclaurin Theorem,

sin x = f(0) + xf '(0) + ( (x^2)/2!) f ''(0) + ( (x^3)/3!) f '''(0) + ...

sin x = x - ( (x^3)/3!) + ( (x^5)/5!)+...+ ((-1)^n) (x^(2n+1))/(2n+1)!)+...