Convergence of the homotopy analysis method

The homotopy analysis method is studied in the present paper. The question of convergence of the homotopy analysis method is resolved. It is proven that under a special constraint the homotopy analysis method does converge to the exact solution of the sought solution of nonlinear ordinary or partial differential equations. An optimal value of the convergence control parameter is given through the square residual error. An error estimate is also provided. Examples, including the Blasius flow, clearly demonstrate why and on what interval the corresponding homotopy series generated by the homotopy analysis method will converge to the exact solution.