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Theorem 5
 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 nthdegree Taylor polynomial of f at a and lim(n>infinity) R_{n}(x) = 0 for xa < R, the f is equal to the sum of its Taylor series on the interval xa < R

