Probabilistic interpretation of Taylor series

In mathematics, the Taylor series is a power series associated to a function. From a probabilistic point of view, the Taylor series is the most natural approximation of that function.

More precisely, the Taylor polynomial of degree $,\; n\; ,$ is the most naturalapproximation of a function $,\; f\; ,$ which is at least$,\; n\; ,$ times differentiable. If $,\; f\; ,$ has this property in a neighborhoodof a point $,\; a\; ,$, we have from the mean value theorem:$,\; f(a+h)=\; f(a)\; +(f(a+h)-f(a))\; =\; f(a)\; +\; f\text{'}(a+v\_1h)\; h\; ,$ where $,\; v\_1\; ,$ is some valuewith $,\; 0\; leq\; v\_1\; leq\; 1.\; ,$ The same argument applied to $,\; f\text{'}(a+v\_1h)\; ,$ yields :$,\; f\text{'}(a+v\_1h)\; =\; f\text{'}(a)+v\_1h\; f"(a+v\_2v\_1\; h)\; ,$ for some $0\; leq\; v\_2\; leq\; 1,$ and, by mathematical induction,:$,\; f^\{(k)\}\; (a+v\_k\; v\_\{k-1\}\; cdots\; v\_1h)\; =\; f^k(a)+v\_k\; v\_\{k-1\}\; cdots\; v\_1\; f^\{(k+1)\}(a+\; v\_\{k+1\}\; cdots\; v\_1\; h)\; h\; ,$ for mean values $,\; v\_1,\; v\_2,\; dots\; ,v\_k,.$

Substituting these terms into $,\; f(a+h)\; ,$ we obtain:$,\; f(a+h)\; =\; f(a)+f\text{'}(a)\; h+v\_1f"(a)\; h^2+v\_1^2\; v\_2\; f"\text{'}(a)\; h^3\; +cdots\; +v\_1^\{n-1\}v\_2^\{n-2\}\; cdots\; v\_\{n-1\}\; f^\{(n)\}(a)\; h^n,$ plus an error term.

Without any information about the mean values $,\; v\_1,\; v\_2,\; dots\; ,$ (other than their existence) it is natural (in the senseof unbiased)to model these values as independent uniform random variables on [0,1] . This turns $,\; f(a+h)\; ,$ into a random variable $,\; ilde\; f(a+h)\; ,$for which the error term tends to zero almost surely as $,\; n\; ,$ tends toinfinity, and its mathematical expectation is exactly theTaylor series! Thus the Taylor polynomial can be considered to be the most natural (unbiased) approximationof such functions.This approach also underlines the important link between Taylor series and the mean value theorem.

References

* Franz Thomas Bruss (1982) A Probabilistic approach to an Approximation Problem. Annales de la Soc. Scientifique de Bruxelles, Vol. 96, II, 91-97.