Definition

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:$$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots$$Or, more compactly: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$$ ### 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.