Asymptotic expansions of several series and their application

Details Asymptotic expansions of several series and their application Asymptotic expansions of several series and their application

2010-01-30

arxiv.org/abs/1002.0083v1

Ukrainian Mathematical Bulletin, vol. 6 (2009), N 4

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or $\epsilon=-1$. These expansions are applied to obtain precise inequalities for Mathieu series. Keywords: Asymptotic expansion, residues, generalized Mathieu series, inequalities.

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