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Keywords:
arithmetic and geometric means; discrete inequality
arithmetic and geometric means; discrete inequality
Summary:
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 $.
References:
[1] Fichtenholz G.M.: Differential - und Integralrechnung, II. Dt. Verlag Wissensch., Berlin, 1979. MR 0238636 | Zbl 0900.26002
[2] Minc H., Sathre L.: Some inequalities involving $(r!)^{1/r}$. Edinburgh Math. Soc. 14 (1964/65), 41-46. MR 0162751
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