| Title:
|
Note on special arithmetic and geometric means (English) |
| Author:
|
Alzer, Horst |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1994 |
| Pages:
|
409-412 |
| . |
| Category:
|
math |
| . |
| 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 $. (English) |
| Keyword:
|
arithmetic and geometric means |
| Keyword:
|
discrete inequality |
| MSC:
|
26A99 |
| MSC:
|
26D15 |
| MSC:
|
26D99 |
| MSC:
|
40A05 |
| idZBL:
|
Zbl 0806.26015 |
| idMR:
|
MR1286588 |
| . |
| Date available:
|
2009-01-08T18:12:02Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118680 |
| . |
| Reference:
|
[1] Fichtenholz G.M.: Differential - und Integralrechnung, II.Dt. Verlag Wissensch., Berlin, 1979. Zbl 0900.26002, MR 0238636 |
| Reference:
|
[2] Minc H., Sathre L.: Some inequalities involving $(r!)^{1/r}$.Edinburgh Math. Soc. 14 (1964/65), 41-46. MR 0162751 |
| . |
