$5\mathrm{.}2\mathrm{.}19$ Show that the following series is convergent.

${\displaystyle \underset{s=0}{\overset{}{}}}\frac{(2s-1)!!}{(2s)!!(2s+1)}$

Note. $(2s-1)!(2s-1)(2s-3)cdots3*1$ with $(-1)!1$;$(2s)!(2s)(2s-2)cdots4*2$ with $0!1.$ The series appears as a series expansion of $si{n}^{-1}(1)$ and equals $\frac{}{2},$ and $si{n}^{-1}x-=arcsinx(sinx{)}^{-1}.$