Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a certain regularity condition on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is super-geometric. Furthermore, at proper Gaussian points or Lobatto points, the rate of convergence may gain one or two orders of the polynomial degree.