Variational eigenvalues for the Rydberg states of helium: Comparison with experiment and with asymptotic expansions

High-precision variational eigenvalues are presented for a range of helium Rydberg states up to n=10 and L=7. Convergence to a few parts in ${10}^{18}$ is obtained for many of the nonrelativistic eigenvalues. The results allow a clear assessment of the accuracy of asymptotic expansion methods extensively developed for states of high angular momentum. After adding relativistic and radiative corrections, a comparison with new high-precision measurements for transitions among the n=10 states is made. Contributions from the long-range Casimir-Polder effect are discussed.