CEU Electronic Theses and Dissertations, 2014
| Author | Issaka, Aziz |
|---|---|
| Title | Some Results on Two Asymptotic Series of Ramanujan |
| Summary | This thesis present 4 result about two problems of Ramanujan. For the first problem, which is about Ramanujan's inverse Digamma approximation. Firstly, we gave an explicit formula for computing the coefficients when $n=1/4$. Secondly, we provided a recurrence relation for the general term of the asymptotic series for every complex number $n$. Finally, we provided an asymptotic formula for the general term. From our final result, we realized that even though the general terms looks very complicated, their asymptotic behaviour are simple which also shows the divergent character of Ramanujan's series. For the second problem, which is about Ramanujan's asymptotic formula for the $n$th Harmonic number. Firstly, we provided an expression for the $\sum_{k=n+1}^{\infty}1/k^j$ where $j>1$ with a formula for the general term and a precise error term. Secondly, we provided an expression for the odd powers, which is in terms of the reciprocal of the $n$th triangular number, with a formula for the general term and a nice error term. This as a result, improved the paper of M. D. Hirschhorn entitled ``Ramanujan's enigmatic formula for the harmonic series''. |
| Supervisor | Boroczky Karoly/Nemes Gergo |
| Department | Mathematics MSc |
| Full text | https://www.etd.ceu.edu/2014/issaka_aziz.pdf |
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