TITLE: Newton's way of getting a Taylor expansion
QUESTION [8 upvotes]: I don't understand how Newton find the Taylor expansion of $\frac{a^2}{b+x}$ by the following method :

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

REPLY [2 votes]: $$\begin{align}&aa\;&\mid& b+x\\&aa+\frac{aa}bx&\mid&\frac{aa}b\\
&------&--&----------\\
&0-\frac{aa}bx&\mid&\frac{aa}b-\frac{aax}{b^2}\\
&\;\;-\frac{aa}bx-\frac{aax^2}{b^2}&\mid&\\&------&--&----------\\&\frac{aax^2}{b^2}&\mid& \end{align}$$
and etc.
he thus gets
$$\frac{aa}{b+x}=\frac{aa}b-\frac{aax}{b^2}+\frac{aax^2}{b^3}-\ldots$$