推广的Wallis数列的单调性

Abstract

讨论了推广的Wallis数列n+c∫π20sinnxdx(n≥1,c为非负常数)的单调性.黄永忠等(2016)证明了当0≤c≤12该数列严格递增;当12

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….