当 $x\in R,\left|x\right|<1$时，有如下表达式： $1+x+x^2+\cdots +x^n+\cdots =\frac{1}{1-x}$两边同时积分得： ${\int }_0^{\frac{1}{2}}1dx+{\int }_0^{\frac{1}{2}}xdx+{\int }_0^{\frac{1}{2}}{x}^{2}dx+\cdots +{\int }_0^{\frac{1}{2}}{x}^{n}dx+\cdots ={\int }_0^{\frac{1}{2}}\frac{1}{1-x}dx$.从而得到如下等式：
$1\times \frac{1}{2}+\frac{1}{2}\times {\left(\frac{1}{2}\right)}^2+\frac{1}{3}\times {\left(\frac{1}{2}\right)}^3+\cdots +\frac{1}{n+1}\times {\left(\frac{1}{2}\right)}^{n+1}+\cdots =\ln 2$.请根据以上材料所蕴含的数学思想方法，计算：
$C_n^0\times \frac{1}{2}+\frac{1}{2}{C}_n^1\times {\left(\frac{1}{2}\right)}^2+\frac{1}{3}{C}_n^2\times {\left(\frac{1}{2}\right)}^3+\cdots +\frac{1}{n+1}{C}_n^n\times {\left(\frac{1}{2}\right)}^{n+1}=$.