Note on special arithmetic and geometric means

Alzer, Horst. "Note on special arithmetic and geometric means." Commentationes Mathematicae Universitatis Carolinae 35.2 (1994): 409-412. <http://eudml.org/doc/247593>.

@article{Alzer1994,
abstract = {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\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty $.},
author = {Alzer, Horst},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {arithmetic and geometric means; discrete inequality; arithmetic mean; geometric mean; inequalities; monotonic sequence; convergence},
language = {eng},
number = {2},
pages = {409-412},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Note on special arithmetic and geometric means},
url = {http://eudml.org/doc/247593},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Alzer, Horst
TI - Note on special arithmetic and geometric means
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 2
SP - 409
EP - 412
AB - 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\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty $.
LA - eng
KW - arithmetic and geometric means; discrete inequality; arithmetic mean; geometric mean; inequalities; monotonic sequence; convergence
UR - http://eudml.org/doc/247593
ER -