推广的Wallis数列的单调性

Monotonicity of Generalized Wallis Sequences

讨论了推广的Wallis数列{(n+c)(1/2)∫π/20sin^nxdx}(n≥1,c为非负常数)的单调性.黄永忠等(2016)证明了当0≤c≤1/2该数列严格递增;当1/2<c≤1该数列对于充分大的n严格递减.本文给出了此结论的一个新的简洁证明,并对相关问题做了讨论.进一步,证明了当且仅当c>2π~2-16/16-π~2=0.609945…,推广的Wallis数列为严格递减数列.

Let yn(c)=n+c∫π20sinn(x)dx(n=1,2,…;c≥0).Huang et.al(2016)proved that the sequence{yn(c)}is strictly increasing if0≤c≤12,and it is strictly decreasing for sufficiently large n if12<c≤1.We give a new proof of this result and give some discussions on related problems.Moreover,we show that the sequence{yn(c)}is strictly decreasing if and only if c>2π2-1616-π2=0.609945….