Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series
Mark B. Villarino
Abstract
A 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.
Create a lesson
Related papers
An equivalence principle for martingales
Grigori A. Karagulyan, Linda A. Khachatryan, Khazhakanush V. Navoyan
Macroscopic return-time spectra and Hausdorff dimension transitions
Jinjun Li
Homogeneous Polynomial Carleson Operators Along the Paraboloid
Alina Zhu
Stein's generalized spherical maximal operator
Luz Roncal
Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets
Yan-Song Fu, Xu-Guang Wang, Meng Zhu
Sampling Density for Multi-window STFT Phase Retrieval
Wenchang Sun, Mengxuan Zhou