2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70240A simple integration by parts and telescopic cancellation leads to a rigorous derivation of the first 2 terms for the error in Ramanujan's asymptotic series for the nth partial sum of the harmonic series. Then Kummer's transformation gives three more terms and a rigorous error estimate. Finally best-possible estimates of Lodge's approximations.7 pages. Numerical copy error corrected as well as the computations based on it. Minor expository improvementsClassical Analysis and ODEs40A25text