## Definition of Taylor series

**:**a power series that gives the expansion of a function

*f*(

*x*) in the neighborhood of a point

*a*provided that in the neighborhood the function is continuous, all its derivatives exist, and the series converges to the function in which case it has the form {latex}f(x) = f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^{2} + \dots + \frac{f^{[n]}(a)}{n!}(x - a)^{n}{/latex} where f[

^{n}] (

*a*) is the derivative of nth order of

*f*(

*x*) evaluated at

*a*— called also Taylor's series