Geodesic
Note on special arithmetic and geometric means
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 409-412

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We prove: If $A(n)$ and $G(n)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA(n)/G(n)-(n-1)A(n-1)/G(n-1)$ $(n\geq 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty $.
Classification : 26A99, 26D15, 26D99, 40A05
Keywords: arithmetic and geometric means; discrete inequality 
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Alzer, Horst. Note on special arithmetic and geometric means. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) no. 2, pp. 409-412. http://geodesic.mathdoc.fr/item/CMUC_1994__35_2_a21/