Ramanujan harmonic numbers expansion

In this paper, we develop a direct formulation based on the variable X=n(n+1)+h, which h is a real parameter. This method starts from the digamma representation of H_n, recenters the expansion at n+1/2, and then converts the result into descending powers of X. This actually gives an explicit coefficient formula in terms of Bernoulli polynomials evaluated at 1/2. The special value h=1/3 is shown to cancel the first correction term, giving a sharper expansion whose error begins at a higher order. Numerical comparisons showed that the shifted form improves substantially over the standard Euler truncation and the unshifted pronic expansion for small and moderate n and this gives transparent route to Ramanujan-type harmonic-number approximations.