The value of $\sum _{n=1}^{\infty }\frac{1}{n{.2}^{n-1}}$ is equal to __________

$f\left(x\right)=1+\frac{x}{2}+\frac{x^2}{4}+\frac{3}{8}+\mathrm{.......}$ 

Note that if we take the integral of f(x) We get the function $F\left(x\right)=x+\frac{x^2}{2.2}+\frac{x^3}{{3.2}^2}+\mathrm{.....}$  

Evaluating this integral in the interval [0, 1]  We get $1+\frac{1}{{2.2}^2}+\frac{1}{{3.2}^2}+\mathrm{......},$ which is the desired sum.