Taylor's Theorem and Newton's Method of Divided Differences

Question

While reading Chandrashrkhar's edition of Principia , I came to know that Newton's Method of Divided Differences can be used to prove Taylor's Theorem. Could some one help me in knowing how this is possible. I found this http://www.sciencedirect.com/science/article/pii/S037704279900360X but it didn't help .

Edit

Note- I want this

Could someone give a reference to a book or a link where I can study the method of divided differences in sense in which Newton developed it

Answer 1

Not only can it be done, it is how Taylor did it: B. Taylor, Methodus incrementorum directa et inversa (1715), 21--23, Prop. VII Theor. III & Coroll. II..

Taylor developed Taylor series through uniformly spaced nodes at $t_j=t_0+j\,\Delta t$, interpolating $x=x(t)$ between $t_0$ (Taylor uses $z$ equally for $t$ and $t_0$) and $t_0+v$, for some increment $v$ in $n$ steps. From Newton's interpolation formula, Taylor derived the formula (Prop. VII) $$ \sum_{k=0}^n {(t-t_0) \strut\over\strut 1}{(t-t_0-\Delta t) \strut\over\strut 2} {(t-t_0-2\Delta t) \strut\over\strut 3}\cdots{(t-t_0-(k-1)\Delta t) \strut\over\strut k} {\Delta^k x \over \Delta t^k}\,. $$ He wrote the proposition in terms of forward differences, but it is, of course, equivalent to divided differences.

Letting $\Delta t \rightarrow 0$ (Si pro Incrementis evanescentibus scribantur fluxiones...), Taylor obtained that ${\Delta^k y / \Delta t^k} \rightarrow {d^ky /dt^k}$ and the other factors approach $(t-t_0)^k/k!$, which yields the Taylor series, since $n$ is determined by $v$ and $\Delta t$. Klein called it a "a limit-taking of unheard-of boldness" ("Hierin liegt nun tatsächlich Grenzübergang von unerhörter Kühnheit"), Elementarmathematik vom höheren Standpunkte aus: Arithmetik, Algebra, Analysis (1908).