If  $\left|x\right|<1,$ then in the expansion  of  ${(1+2x+3x^2+4x^3+\mathrm{....})}^{1/2}$,  the coefficient  of  $x^n$ is 

Since $1+2x+3x^2+4x^3+\mathrm{....}\infty ={(1-x)}^{-2}$
There fore ,we have  ${(1+2x+3x^2+4x^3+\mathrm{....}\infty )}^{1/2}={\left\{{(1-x)}^{-2}\right\}}^{1/2}$
$={(1-x)}^{-1}=1+x+x^2+\mathrm{....}+x^n+\mathrm{.....}\infty$