# State and prove the taylar'stheorem?

Relevance
• You might have meant how in general is a Taylor series derived

(when about x = 0 it is called the Maclaurin series)

One method uses repeated differentiation as in this example for sin(x)

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• wikipedia:

Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a given point are finite-order truncations of its Taylor series, which completely determines the function in some neighborhood of the point. It can be thought of as the extension of linear approximation to higher order polynomials, and in the case of k equals 2 is often referred to as a quadratic approximation. The exact content of "Taylor's theorem" is not universally agreed upon. Indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial.

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