Abstract
In this paper, we provide a method for constructing a continued fraction approximation based on a given asymptotic expansion. We establish some asymptotic expansions for the harmonic numbers which employ the nth triangular number. Based on these expansions, we derive the corresponding continued fraction approximations for the harmonic numbers.
Highlights
The Indian mathematician Ramanujan claimed the following asymptotic expansion for the nth harmonic number: n1 1Hn := k ∼ 2 ln(2m) + γ + 12m − 120m2 + 630m3 − 1680m4 + 2310m5 k=1 (1.1)− 360360m6 + 30030m7 − 1166880m8 + 17459442m9 − · · ·as n → ∞, where m = n(n + 1)/2 is the nth triangular number and γ is the Euler-Mascheroni constant.Chao-Ping Chen and Qin WangRamanujan’s formula (1.1) is the subject of intense investigations and has motivated a large number of research papers.Villarino [12, Theorem 1.1] first gave a complete proof of expansion (1.1) in terms of the Bernoulli polynomials
We provide a method to construct a continued fraction approximation based on a given asymptotic expansion
We establish some asymptotic expansions for the harmonic numbers which employ the nth triangular number
Summary
We provide a method for constructing a continued fraction approximation based on a given asymptotic expansion. We establish some asymptotic expansions for the harmonic numbers which employ the nth triangular number. Based on these expansions, we derive the corresponding continued fraction approximations for the harmonic numbers
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