IJPAM: Volume 38, No. 4 (2007)

THE MONOTONICITY OF A NUMBER SEQUENCE
INVOLVING THE VOLUME OF THE UNIT BALL IN $R^n$

Fangli Xia$^1$, Yuming Chu$^2$, Gendi Wang$^3$
$^1$Department of Mathematics
Hunan City University
Hunan, Yiyang, 413000, P.R. CHINA
$^{2,3}$Department of Mathematics
Huzhou Teachers College
Zhejiang, Huzhou, 313000, P.R. CHINA
$^2$e-mail: chuyuming@hutc.zj.cn


Abstract.Let $\Omega_n=\pi^{
n/2}/\Gamma(1+\frac n2)$ be the volume of the unit ball in $R^n$, and $x_n=\frac{(n+1)^{\alpha}\Omega_{n+1}}{\Omega_n}$ $(n=1,2,3\cdots)$ be a number sequence. In this paper, we prove that $x_n$ is decreasing for $\alpha\leq \frac{8}{21}=0.380952\cdots$ and increasing for $\alpha\geq \frac{31}{60}=0.516666\cdots$.

Received: June 17, 2007

AMS Subject Classification: 33B15

Key Words and Phrases: gamma function, monotonicity, volume

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 38
Issue: 4