= Taylor's theorem
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= Taylor theorem
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<Taylor's theorem> expresses a sufficiently <differentiable function> as a finite <Taylor polynomial> plus a <Taylor remainder>. One convenient version assumes $f$ is $n+1$ times continuously <differentiable> on the segment between $a$ and $x$. Then
$$
f(x)=T_{n,a}(x)+R_{n,a}(x),\qquad R_{n,a}(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}
$$
for some intermediate point $\xi$, when $x\ne a$. Thus a bound $|f^{(n+1)}|\leq M$ on that segment gives $|R_{n,a}(x)|\leq M|x-a|^{n+1}/(n+1)!$. The intermediate point usually need not be computed: bounding the <derivative> throughout the segment is enough. This is why local approximation can yield reliable numerical bounds. A finite <Taylor polynomial> does not require an infinite <Taylor series> to represent the <function>.
Historically, the Royal Society holds <Brook Taylor>'s letter of 26 July 1712 to John Machin. Its catalogue connects his work on <Kepler's second law> with the expansion later published in the 1715 Methodus incrementorum directa et inversa. This is a documented route from an orbital calculation to a general approximation method, rather than a claim that every earlier use of such expansions began with Taylor. See https://catalogues.royalsociety.org/calmview/Record.aspx?id=MS%2F82%2F2&src=CalmView.Catalog[Royal Society archive, letter MS/82/2].
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