information:highlighting, structural:reformatting
Definition

The Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the power series f (a) + f ′ (a) 1 ! (x − a) + f ″ (a) 2 ! (x − a) 2 + f ‴ (a) 3 ! (x − a) 3 + ⋯ = ∑ n = 0 ∞ f (n) (a) n ! (x − a) n. {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}.} Here, n ! denotes the factorial of n. The function f (a) denotes the n th derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and (x − a) and 0! are both defined to be 1. This series can be written by using sigma notation, as in the right side formula. With a = 0, the Maclaurin series takes the form: f (0) + f ′ (0) 1 ! x + f ″ (0) 2 ! x 2 + f ‴ (0) 3 ! x 3 + ⋯ = ∑ n = 0 ∞ f (n) (0) n ! x n. {\displaystyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+{\frac {f'''(0)}{3!}}x^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}.}