- Taylor Series Expansion

Introduction to Taylor Series #

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function’s derivatives at a single point. It provides a way to approximate functions using polynomial expressions.

Basic Formula #

The Taylor series expansion of a function $f(x)$ around a point $x = a$ is given by:

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

Where:

• $f^{(n)}(a)$ represents the $n$-th derivative of $f$ evaluated at point $a$
• $(x-a)$ is the distance from the expansion point

Alternative Notation #

If we denote $x-a = \Delta x$, the Taylor series can be written as:

$$f(a+\Delta x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(\Delta x)^n$$

Shifting the Expansion Point #

When we shift the expansion point from $a$ to $x$, the formula becomes:

$$f(x+\Delta x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(x)}{n!}(\Delta x)^n$$

This form is particularly useful for local approximations of functions around a variable point $x$.