If $\left|\mathrm{x}\right|<\frac{1}{2},$ then the coefficient of ${\mathrm{x}}^{\mathrm{r}}$ in the expansion of $\frac{1+2\mathrm{x}}{{(1-2\mathrm{x})}^2}$ is

$\begin{array}{rcl}\frac{1+2\mathrm{x}}{{(1-2\mathrm{x})}^2}& =& (1+2\mathrm{x}) {(1-2\mathrm{x})}^{-2} \\ & =& (1+2\mathrm{x}) (1+2(2\mathrm{x})+3{\left(2\mathrm{x}\right)}^2) \\ & & +.....+\mathrm{r}{\left(2\mathrm{x}\right)}^{\mathrm{r}-1}+(\mathrm{r}+1) {\left(2\mathrm{x}\right)}^{\mathrm{r}} \\ & & +.....)\end{array}$

Coeff of ${\mathrm{x}}^{\mathrm{r}}=(\mathrm{r}+1) 2^{\mathrm{r}}+2 \mathrm{r} 2^{\mathrm{r}-1}$

                   $=2^{\mathrm{r}} (2\mathrm{r}+1)$