Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 4 / b / Solution

Use the Taylor expansion $\sqrt{1-x}=1-x/2-x^2/8+O(x^3)$. Expanding the rational form givesMatching through degree two yields $a_0=1$, $a_1-b_1=-1/2$, and $b_1^2-a_1{b}_1=-1/8$. The last equation becomes $b_1/2=-1/8$, soThis is the $[1/1]$ Padé approximant about zero. Its pole is at four, outside the interpolation interval; its order of matching is local and does not assert uniform minimax optimality.