## Can one do better than linear convergence?

__Duh__!## Expand

*f*(*x*) in a Taylor series:*f*(*x*+ ) =*f*(*x*) +*f '*(*x*) +*f "*(*x*) 2/2 + ...

## For <<

*x*, drop higher order terms, so:*f*(*x*+ ) = 0 = -*f*(*x*) /*f '*(*x*)

## is correction added to current guess of root:

- i.e.
*xi*+1 =*xi*+

- i.e.