On a sequence transformation with integral coefficients for Euler’s constant
Author:
C. Elsner
Journal:
Proc. Amer. Math. Soc. 123 (1995), 15371541
MSC:
Primary 11Y60; Secondary 40A05, 65B05
DOI:
https://doi.org/10.1090/S00029939199512339694
MathSciNet review:
1233969
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Abstract: Let $\gamma$ denote Euler’s constant, and let \[ {s_n} = \left ( {1 + \frac {1}{2} + \cdots + \frac {1}{{n  1}}} \right )  \log n\quad (n \geq 2).\] 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 \[ {\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\mu _{n,n}}{s_{\tau + n}} = \gamma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{  1}}),\] 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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Article copyright:
© Copyright 1995
American Mathematical Society