Taylor theorem

The Taylor theorem expresses a sufficiently differentiable function as a finite Taylor polynomial plus a remainder.

For a continuously differentiable vector-valued function whose domain contains the segment from $x$ to $x+h$, apply the fundamental theorem of calculus componentwise to $t\mapsto F(x+th)$. The resulting integral matrix is valid even when there is no common intermediate point giving a vector mean-value formula. For a score function, it integrates the Hessian matrix along the segment to a candidate estimate.

If $f$ has $k+1$ continuous derivatives, its Taylor remainder about $a$ is

A Taylor expansion about a point consists of successive derivative terms there, together with either a finite remainder or an infinite Taylor series when that series represents the function.

A Peano zero through order $r$ means that the function admits the zero Taylor polynomial with remainder $o(|t{|}^r)$ at $x_0$. This does not assert the existence of ordinary iterated derivatives on a neighborhood. For example, multiplying $t^{r+1}$ by a bounded continuous function with dense points of nondifferentiability preserves this estimate while destroying neighborhood differentiability.

A parabolic approximation retains terms through second order in a local Taylor expansion. Near the closest point of a circle of radius $a$ and its tangent line, the separation is $x^2/(2a)+O(x^4/a^3)$.

For a polynomial $p$ of degree at most $m$, the Taylor formula is exact:

If $f$ is $n$ times differentiable between $a$ and $x$, then for some $\xi$ strictly between them,

The Taylor remainder is the difference between a function and a finite Taylor polynomial. Bounds on higher derivatives or their Hölder continuity control this error.

If $f^{(m-1)}$ is locally absolutely continuous, then its Taylor remainder can be written as an integral involving the weak derivative $f^{(m)}$. This form permits norm bounds when the highest derivative is only integrable.

The Taylor series of a sufficiently regular function about $a$ is