Editing Taylor series (section)

===Generalization===
The generalization of the Taylor series does converge to the value of the function itself for any [[bounded function|bounded]] [[continuous function]] on {{math|(0,∞)}}, and this can be done by using the calculus of [[finite differences]]. Specifically, the following theorem, due to [[Einar Hille]], that for any {{math|''t'' > 0}},<ref>{{multiref
 |{{harvnb|Feller|2003|p=230&ndash;232}}
 |{{harvnb|Hille|Phillips|1957|pp=300–327}}
}}</ref>

<math display="block" >\lim_{h\to 0^+}\sum_{n=0}^\infty \frac{t^n}{n!}\frac{\Delta_h^nf(a)}{h^n} = f(a+t).</math>

Here {{math|Δ{{su|p=''n''|b=''h''}}}} is the {{mvar|n}}th finite difference operator with step size {{mvar|h}}. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the [[Newton series]]. When the function {{mvar|f}} is analytic at {{mvar|a}}, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

In general, for any infinite sequence {{math|''a''<sub>''i''</sub>}}, the following power series identity holds:

<math display="block">\sum_{n=0}^\infty\frac{u^n}{n!}\Delta^na_i = e^{-u}\sum_{j=0}^\infty\frac{u^j}{j!}a_{i+j}.</math>

So in particular,

<math display="block">f(a+t) = \lim_{h\to 0^+} e^{-t/h}\sum_{j=0}^\infty f(a+jh) \frac{(t/h)^j}{j!}.</math>

The series on the right is the [[expected value]] of {{math|''f''&thinsp;(''a'' + ''X'')}}, where {{mvar|X}} is a [[Poisson distribution|Poisson-distributed]] [[random variable]] that takes the value {{math|''jh''}} with probability {{math|''e''<sup>−''t''/''h''</sup>·{{sfrac|(''t''/''h''){{isup|''j''}}|''j''!}}}}. Hence,

<math display="block">f(a+t) = \lim_{h\to 0^+} \int_{-\infty}^\infty f(a+x)dP_{t/h,h}(x).</math>

The [[law of large numbers]] implies that the identity holds.{{sfn|Feller|2003|p=231}}