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I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method :

Newton's Method

**This screenshot is from : The method of fluxions and infinite series

Any idea ?

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    $\begingroup$ ...and this is yet another reason why to read originals of old mathematicians' works can be pretty annoying and frustrating: their notation, among other things, is close to impossible to understand if we don't first learn what they meant. My advice is to try to get a version of the original but with (1) modern notation, and (2) explaining notes of what the author meant in some steps, what he knew, what he didn't, etc. Interesting question. +1 $\endgroup$ – Timbuc Oct 24 '14 at 23:31
  • $\begingroup$ He is dividing $a^2$ by $b+x$ using long division, and his method is the same as the one we use today, except he is putting the quotient to the right of the $a^2$ instead of on top of it, as we do now. $\endgroup$ – user84413 Oct 24 '14 at 23:40
  • $\begingroup$ This is just high-school long division where you write the terms in ascending degree instead of descending. It works very nicely for dividing any power series into another. $\endgroup$ – Lubin Oct 25 '14 at 2:51
  • $\begingroup$ The fraction ... being propofed $\endgroup$ – Pedro Tamaroff Nov 15 '14 at 15:31
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$$\begin{align}&aa\;&\mid& b+x\\&aa+\frac{aa}bx&\mid&\frac{aa}b\\ &------&--&----------\\ &0-\frac{aa}bx&\mid&\frac{aa}b-\frac{aax}{b^2}\\ &\;\;-\frac{aa}bx-\frac{aax^2}{b^2}&\mid&\\&------&--&----------\\&\frac{aax^2}{b^2}&\mid& \end{align}$$

and etc.

he thus gets

$$\frac{aa}{b+x}=\frac{aa}b-\frac{aax}{b^2}+\frac{aax^2}{b^3}-\ldots$$

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