Taylor Series

Last updated Nov 28, 2022

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.

${\displaystyle f(a)+{\frac {f’(a)}{1!}}(x-a)+{\frac {f’’(a)}{2!}}(x-a)^{2}+{\frac {f’’’(a)}{3!}}(x-a)^{3}+\cdots }$

where n! denotes the factorial of n. In the more compact sigma notation, this can be written as

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}}!$

where f(n)(a) denotes the nth derivative of f evaluated at point a.