[Paper Review] Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number

Mark B. Villarino|ArXiv.org|Jul 26, 2007

Advanced Mathematical Identities3 references14 citations

TL;DR

This paper provides a complete derivation and error analysis of Ramanujan's asymptotic expansion for the $n$th harmonic number $H_n$ in negative powers of the $n$th triangular number $m = n(n+1)/2$. It establishes a general formula for the coefficients $R_p$ in the expansion $H_n = \frac{1}{2}\ln(2m) + \gamma + \sum_{p=1}^r \frac{R_p}{m^p} + \Theta_r \frac{R_{r+1}}{m^{r+1}}$, with $0 < \Theta_r < 1$, proving the series is asymptotic and offering sharp error bounds.

ABSTRACT

An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.

Motivation & Objective

• To provide a complete, rigorous derivation of Ramanujan's asymptotic expansion for the $n$th harmonic number $H_n$ in terms of negative powers of the $n$th triangular number $m = n(n+1)/2$.
• To establish a general closed-form formula for the coefficients $R_p$ in Ramanujan's expansion, resolving a gap left by Berndt's verification-only proof.
• To prove that Ramanujan's series is asymptotic by deriving a sharp error estimate of the form $\Theta_r \cdot \frac{R_{r+1}}{m^{r+1}}$ with $0 < \Theta_r < 1$.
• To compare Ramanujan's approximation with other known approximations (e.g., DeTemple–Wang, Lodge) and analyze their relative accuracy.
• To trace the historical development of the formula, identifying Cesàro (1885) as the earliest known appearance and Lodge (1904) as the first to provide error bounds.

Proposed method

• Derives the general coefficient formula $R_p = \frac{(-1)^{p-1}}{2p \cdot 8^p} \left\{ 1 + \sum_{k=1}^p \binom{p}{k} (-4)^k B_{2k}(1/2) \right\}$ using algebraic transformations of the DeTemple–Wang half-integer approximation.
• Applies the Euler–Maclaurin summation formula as a foundation, connecting Ramanujan's expansion to the standard asymptotic expansion of $H_n$ in inverse powers of $n$.
• Uses generating function techniques and properties of Bernoulli polynomials $B_{2k}(1/2)$ to derive and simplify the coefficient expression.
• Employs alternating series error estimation principles to bound the remainder term, showing that the error is less than the next term in absolute value and of the same sign.
• Introduces auxiliary quantities $\epsilon_r$, $E_r$, and $\theta_r$ to decompose the total error and proves the existence of $\Theta_r \in (0,1)$ such that the error term is $\Theta_r \cdot R_{r+1}/m^{r+1}$.
• Validates the asymptotic nature of the series by proving that truncating after $r$ terms yields an error bounded by the magnitude of the next term, confirming the series is asymptotic.

Experimental results

Research questions

• RQ1 What is the general closed-form expression for the coefficients in Ramanujan's expansion of $H_n$ into negative powers of the $n$th triangular number?
• RQ2 Is Ramanujan's series asymptotic, and can a sharp error estimate be rigorously proven?
• RQ3 How does Ramanujan's approximation compare in accuracy to other known approximations such as DeTemple–Wang and Lodge’s formulas?
• RQ4 What is the historical origin of Ramanujan’s formula, and who first established related results?
• RQ5 Can the method used for $H_n$ be adapted to derive a new Stirling-type expansion for $n!$ in powers of the triangular number $m$?

Key findings

• The general coefficient in Ramanujan’s expansion is given by $R_p = \frac{(-1)^{p-1}}{2p \cdot 8^p} \left\{ 1 + \sum_{k=1}^p \binom{p}{k} (-4)^k B_{2k}(1/2) \right\}$, providing a complete analytical formula for all terms.
• The expansion $H_n = \frac{1}{2}\ln(2m) + \gamma + \sum_{p=1}^r \frac{R_p}{m^p} + \Theta_r \cdot \frac{R_{r+1}}{m^{r+1}}$ is asymptotic, with $0 < \Theta_r < 1$, confirming the error is bounded by the next term in magnitude and sign.
• The error term is shown to be of the form $\Theta_r \cdot R_{r+1}/m^{r+1}$, with $\Theta_r \in (0,1)$, proving the series is asymptotic and providing a sharp error estimate.
• The paper traces the history and identifies Cesàro (1885) as the first to state a two-term version of the expansion with an error term, predating Ramanujan.
• Lodge (1904) previously derived similar approximations with best-possible constants in the error terms, but without the general coefficient formula or full asymptotic analysis.
• The method used to derive the expansion can be extended to potentially yield a new Stirling-type expansion for $n!$ in powers of the triangular number $m$, as suggested in the conclusion.