原式$=\large\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\frac{4}{2^4}+\frac{5}{2^5}+.....$

       $=\large\frac{1}{2}+\color{red}{{\frac{1}{2^2}}}+\color{blue}{\frac{1}{2^3}}+\color{teal}{\frac{1}{2^4}}+\color{purple}{\frac{1}{2^5}}+.....$

         $\large\quad\quad+\color{red}{\frac{1}{2^2}}+\color{blue}{\frac{1}{2^3}}+\color{teal}{\frac{1}{2^4}}+\color{purple}{\frac{1}{2^5}}+.....$ 

        $\large\quad\qquad\qquad+\color{blue}{\frac{1}{2^3}}+\color{teal}{\frac{1}{2^4}}+\color{purple}{\frac{1}{2^5}}+.....$

       $\large\qquad\qquad\qquad\qquad+\color{teal}{\frac{1}{2^4}}+\color{purple}{\frac{1}{2^5}}+.....$

       $\large\qquad\qquad\qquad\qquad\qquad\qquad+ \color{purple}{\frac{5}{2^5}}+.....$                             

                                                            $\large+.....$

 $=\large\frac{\frac{1}{2}}{{1-\frac{1}{2}}}+\frac{\frac{1}{2^2}}{{1-\frac{1}{2}}}+\frac{\frac{1}{2^3}}{{1-\frac{1}{2}}}+\frac{\frac{1}{2^4}}{{1-\frac{1}{2}}}+.....$

 $=1\large+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+.....$

 $=\Large\frac{1}{1-\frac{1}{2}}$= 2