abstract:In calculus, 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.
The firstthingyouhave torealize about provingTaylor's theoremis thatthere areinfinitelymanyversions of Taylor's theorem: one for eachpossibleexpressionoftheremainderterm.