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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On a sequence transformation with integral coefficients for Euler's constant


Author: C. Elsner
Journal: Proc. Amer. Math. Soc. 123 (1995), 1537-1541
MSC: Primary 11Y60; Secondary 40A05, 65B05
MathSciNet review: 1233969
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Abstract: Let $ \gamma $ denote Euler's constant, and let

$\displaystyle {s_n} = \left( {1 + \frac{1}{2} + \cdots + \frac{1}{{n - 1}}} \right) - \log n\quad (n \geq 2).$

We prove by Ser's formula for the remainder $ \gamma - {s_n}$ that for all integers $ n \geq 1$ and $ \tau \geq 2$ there are integers $ {\mu _{n,0,}}{\mu _{n,1}}, \ldots ,{\mu _{n,n}}$ such that

$\displaystyle {\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\m... ...mma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{ - 1}}),$

where the constant in $ {O_\tau }$ depends only on $ \tau $.

The coefficients $ {\mu _{n,k}}$ are explicitly given and are bounded by $ {2^{3n + \tau - 1}}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1233969-4
PII: S 0002-9939(1995)1233969-4
Article copyright: © Copyright 1995 American Mathematical Society