  When Newton's method, or Halley's method is used to approximate the $p${th}
root of $1-z$, a sequence of rational functions is obtained. In this paper, a
beautiful formula for these rational functions is proved in the square root
case, using an interesting link to Chebyshev's polynomials. It allows the
determination of the sign of the coefficients of the power series expansion of
these rational functions. This answers positively the square root case of a
proposed conjecture by Guo(2010).
