The Taylor expansion of the exponential function is a powerful mathematical tool that allows us to approximate the exponential function using a polynomial. It is derived using Taylor’s series, a fundamental concept in calculus that enables the expression of a function as an infinite sum of terms, each involving the function’s derivatives evaluated at a specific point. By truncating the series at a finite number of terms, we obtain an approximation of the exponential function that is accurate within a specified error bound. The exponential function’s remarkable property is that its derivatives are all equal to the function itself, leading to a simple and elegant Taylor expansion.
