Newton's Method convergence to the closest $x$? When applying Newton's algorithm, we need to pick an initial guess $x_0$ to start the process.
My question is ... if there is convergence to a root, is this always the root that was closest to $x_0$? 

No.
Let $f(x)=x(x-1)(x+1)$. Newton's method starting at $x_0=0.55$ will converge to $-1$ but the root closest to $x_0$ is $+1$.
The basins of attractions for each root are likely to be complicated, as in the 
Newton fractal.
