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!
Formula 6 - Taylor series of the function f at a
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 }+ ...
Formula 7 - Taylor series of the function f at 0 (Maclaurin series)
f(x) = ( f^{(n)}(0) / n! ) x^{n}
= f(0) + ( f ' (0) / 1! ) x + ( f '' (0) / 2! ) ( x)^{2} + ( f ''' (0) / 3! ) ( x)^{3} + ...
Theorem 8
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