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The Taylor series of a function \(f\) centered at a point \(a\) is given by the following formula: \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\] , where \(f^{(n)}(a)\) refers to the \(n\)-th derivative of the function \(f\) evaluated at the point \(a\), and \(n!\) is the factorial of \(n\). The Taylor series is an infinite sum and represents an approximation of the function \(f\) in terms of its derivatives at a specific point \(a\). It is also referred to as the Taylor polynomial of degree \(n\).

In order for a function \(f\) to have a Taylor series centered at a point \(a\), the following conditions must be satisfied: 1. Differentiability: The function \(f\) must be infinitely differentiable at the point \(a\). This means that the function and all of its derivatives, up to the infinity order, must exist at \(a\). Formally, we can write this as \(f^{(n)}(a)\) existing for all \(n \geq 0\). 2. Convergence: The Taylor series for the function \(f\) must converge to the actual function \(f\). This means that the limit of the partial sum of the Taylor series must equal the function \(f\) at each point in the interval of convergence. Formally, we can write this as: \[\lim_{n\to\infty} \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k = f(x)\] for all \(x\) in the interval of convergence. 3. Smoothness: The function \(f\) must be smooth in the interval of convergence. This means that there can't be any discontinuities or abrupt changes in the function's derivatives in that interval. In summary, for a function \(f\) to have a Taylor series centered at \(a\), it must be infinitely differentiable, have a convergent Taylor series, and be smooth in its interval of convergence.