91Ó°ÊÓ

A Taylor series is a representation of a function as an infinite sum of terms, where each term is based on the function's derivatives at a single point \(a\). The Taylor series for a function \(f\) centered at point \(a\) is given by: \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n\) where \(f^{(n)}(a)\) denotes the \(n^{th}\) derivative of \(f\) evaluated at point \(a\).

In order for a function to have a Taylor series centered at point \(a\), it must satisfy the following conditions: 1. The function \(f\) must be infinitely differentiable at point \(a\), which means that all its derivatives \(\{f'(a), f''(a), f^{(3)}(a), \dots\}\) of \(f\) should exist at \(a\). 2. The function \(f\) must satisfy the Taylor's remainder theorem: for any \(x\) in the interval of convergence, the remainder \(R_n(x)\) of the Taylor series (which is the difference between the actual function and its approximation by the Taylor series) approaches zero as the degree \(n\) goes to infinity: \(\lim_{n \to \infty} R_n(x) = 0\) These are the conditions that must be satisfied by a function \(f\) to have a Taylor series centered at \(a\).