We can find an expression for the kth error
$\epsilon_k$
of the Newton Raphson method, which estimates a root
$\alpha$
of the equation
$f(x)=0$
using the iteration formula
$x_{k+1}=x_k- \frac{f(x_k)}{f'(x_k)}$
.
We can write this as
$x_{k+1}-x_k=- \frac{f(x_k)}{f'(x_k)}$
(1)
Write
$x_{k+1}-\alpha=x_k-\alpha+(x_{k+1}-x_k)$

$\epsilon_{k+1}=\epsilon_k+(x_{k+1}-x_k)$

Into this equation substitute (1) to give
$\epsilon_{k+1}=\epsilon_k- \frac{f(x_k)}{f'(x_k)}$

Now expand numerator and denominator of the second term in Taylor series about
$\alpha$
.
\begin{aligned} \epsilon_{k+1} &= \epsilon_k- \frac{f(\alpha)+f'(\alpha)(x_k-\alpha)+\frac{1}{2}f''(\alpha)(x_k-\alpha)^2+ ...}{f'(\alpha)+f''(\alpha)(x_k-\alpha)+\frac{1}{2}f''''(\alpha)(x_k-\alpha)^2+ ...} \\ & \simeq \epsilon_k- \frac{f'(\alpha)\epsilon_k+\frac{1}{2}f''(\alpha) \epsilon_k^2}{f'(\alpha)+f''(\alpha)\epsilon_k+\frac{1}{2}f''''(\alpha)\epsilon_k^2} \\ &= \epsilon_k- \epsilon_k \frac{f'(\alpha)+\frac{1}{2}f''(\alpha) \epsilon_k}{f'(\alpha)+f''(\alpha)\epsilon_k+\frac{1}{2}f''''(\alpha)\epsilon_k^2} \\ &\simeq \epsilon_k- \epsilon_k \frac{f'(\alpha)+\frac{1}{2}f''(\alpha) \epsilon_k}{f'(\alpha)+f''(\alpha)\epsilon_k+\frac{1}{2}f''''(\alpha)\epsilon_k^2} \\ &= \epsilon_k - \frac{f''(\alpha)}{2f'(\alpha)} \epsilon_k^2 \end{aligned}