Recently I had been reading Elementary Algebra by Hall and Knight, I came upon the division method to find square roots of any multinomial. Out of curiosity, I applied the method to find the root of $1+x$ and arrived at the continued expression$$\sqrt {1+x}=1+x/2-x^2/8+x^3/18-5x^4/128...$$ I plugged in $x=1$ hence arrived at $\sqrt2=1+1/2-1/8..$. A quick search revealed that this is exactly the Taylor expansion of $\sqrt {1+x}$ plugged in with $x=1$.
I am unable to make the connection as to how the Division method yielded a Taylor expansion. Hence can this method produce Taylor expansions of any multinomial?