# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## On a sequence transformation with integral coefficients for Euler’s constantHTML articles powered by AMS MathViewer

by C. Elsner
Proc. Amer. Math. Soc. 123 (1995), 1537-1541 Request permission

## 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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