TaylorAndMaclaurin

• If f has a power series representation (expansion) at a, that is, if: f(x) = c_n(x - a)ⁿ | x - a| < R THEN its coefficients are given by the formula: c_n = f⁽ⁿ⁾(a) / n!

• f(x) = ( f⁽ⁿ⁾(a) / n! ) (x - a)ⁿ
• = f(a) + ( f ' (a) / 1! ) ( x - a) + ( f '' (a) / 2! ) ( x - a)² + ( f ''' (a) / 3! ) ( x - a)³ + ...

• f(x) = ( f⁽ⁿ⁾(0) / n! ) xⁿ
• = f(0) + ( f ' (0) / 1! ) x + ( f '' (0) / 2! ) ( x)² + ( f ''' (0) / 3! ) ( x)³ + ...

if f(x) = T_n(x) + R_n(x), where T_n is the nth-degree Taylor polynomial of f at a and lim(n->infinity) R_n(x) = 0 for |x-a| < R, the f is equal to the sum of its Taylor series on the interval |x-a| < R