问题

令

\[P_n=\frac{(2n-1)!!}{(2n)!!}\]

(1) 对任意 $n>1$, 有

\[\frac{1}{2\sqrt{n}}<\frac{1}{\sqrt{\pi(n+\frac{4}{\pi}-1)}}<P_n<\frac{1}{\sqrt{\pi(n+\frac{1}{4})}}<\frac{1}{\sqrt{3n+1}}<\frac{1}{\sqrt{2n+1}}<\frac{1}{\sqrt{2n}}\]

例如:

\[\frac{1}{15}<\frac{99!!}{100!!}<\frac{1}{10}\]

(2)

\[\frac{(2n)!!}{(2n+1)!!}\frac{2}{\pi}<P_n<\frac{(2n-2)!!}{(2n-1)!!}\frac{2}{\pi}\]

References

匡继昌, 常用不等式