The p-Version of the finite element method has been previously analyzed for elliptic problems with homogeneous boundary conditions. For a homogeneous condition of the Dirichlet type, it was shown that the exponential asymptotic convergence rate was optimal up to an arbitrarily small positive parameter epsilon. In this paper, an alternate proof is discussed which yields a better estimate by removing the dependence on epsilon. The analysis is extended to treat problems with inhomogeneous boundary conditions of both the Dirichlet and Neumann type. Estimates for a case when the solution has singularities at the corners of the domain are also provided. Keywords: Approximation(Mathematics); Polynomials.