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 $\overline{x}$ where $\overline{x}$ is an approximation to the root $p$ such that $f'(\overline{x})\neq 0$ and $|p-\overline{x}|$ is small.

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

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

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

Thanks!