From an article (One more correction formula, Ralph W Snyder) there is a rearrangement of a Taylor series that then clearly shows the Newton Raphson and Halleys formula, I am trying to do this however I cannot make the jump between (1) and (2) (Shown in picture)
Any help would be appreciated.
There is a missing $h$ in the second term of the RHS of (1) and a $+\cdots$ at the end of the RHS of (1), since the RHS of (1) is an infinite series in general. If you take the reciprocal of both sides of the corrected (1), you obtain $$ - \frac{{a_1 h}}{{a_0 }} = \cfrac{1}{{1 + \cfrac{{a_2 }}{{a_1 }}h + \cfrac{{a_3 }}{{a_1 }}h^2 + \cdots }} = 1 - \frac{{a_2 }}{{a_1 }}h + \left( {\left( {\frac{{a_2 }}{{a_1 }}} \right)^2 - \frac{{a_3 }}{{a_1 }}} \right)h^2 - \cdots . $$ Multiplying through by $a_0$, we get $$ - a_1 h = a_0 - \frac{{a_0 a_2 }}{{a_1 }}h + a_0 \left( {\left( {\frac{{a_2 }}{{a_1 }}} \right)^2 - \frac{{a_3 }}{{a_1 }}} \right)h^2 - \cdots . $$ Now add $a_1 h$ to each side and subtract $a_0$ from both sides to obtain the desired result. Note that the term corresponding to $h^2$ in (2) is also written incorrectly (one of the $a_2$ should be $a_3$).