Definition (Taylor expansion)
The Taylor expansion of an arbitrary function f(x) is a way to approximate the function by a series of polynomials, centred around a point a .The expansion expresses f(x) as an infinite sum of its derivatives at a ,weighted by powers of (x−a). The general form is:f(x)=f(a)+f′(a)(x−a)+2!f′′(a)(x−a)2+3!f(3)(a)(x−a)3+⋯Or, more compactly: f(x)=n=0∑∞n!f(n)(a)(x−a)n
Key Points:
- The Taylor series is centred at a, which is called the expansion point.
- The series may converge to the function for all x, or only within some radius of convergence, depending on the function.
- For a=0, the expansion is called a Maclaurin series.
The Taylor expansion is particularly useful when approximating a smooth function near a point, as only the lower-order terms may be needed for a good approximation.