Deriving Newton's root-finding method with Taylor polynomials.

Hi,

when deriving Newton's method using Taylor polynomials my book considers the first Taylor polynomial expanded about $\displaystyle \overline{x}$ where $\displaystyle \overline{x}$ is an approximation to the root $\displaystyle p$ such that $\displaystyle f'(\overline{x})\neq 0$ and $\displaystyle |p-\overline{x}|$ is small.

$\displaystyle f(x) = f(\overline{x}) + (x-\overline{x})f'(\overline{x}) + \frac{(x-\overline{x})^2}{2}f''(\xi(x)),$

where $\displaystyle \xi(x)$ lies between $\displaystyle x$ and $\displaystyle \overline{x}$.

To obtain Newton's method from this, I need to assume $\displaystyle |p-\overline{x}|$ small such that $\displaystyle (p-\overline{x})^2$ is much smaller so that I can ignore it. Does this mean that my initial guess $\displaystyle \overline{x}$ has to be so close to the root p that $\displaystyle 0<|p-\overline{x}|<1$ ?

Thanks!