$\frac{3.5}{5.10}+\frac{\mathrm{3.5.7}}{\mathrm{5.10.15}}+\frac{\mathrm{3.5.7.9}}{\mathrm{5.10.15.20}}+\cdots +\infty =$

$\begin{array}{l}S=\frac{3.5}{2!}{\left(\frac{1}{5}\right)}^2+\frac{3\cdot 5\cdot 7}{3!}{\left(\frac{1}{5}\right)}^3+\frac{3\cdot 5\cdot 7\cdot 9}{4!}{\left(\frac{1}{5}\right)}^4+\cdots \infty \\ 1+\frac{3}{1!}\frac{1}{5}+s=1+\frac{3}{1!}\frac{1}{5}+\frac{3.5}{2!}{\left(\frac{1}{5}\right)}^2+\frac{3\cdot 5\cdot 7}{3!}{\left(\frac{1}{5}\right)}^3+\cdots +\infty \\ Herep=3\\ p+q=5\\ q=2\\ \frac{x}{q}=\frac{1}{5}\\ x=\frac{2}{5}\\ \frac{8}{5}+S={(1-\frac{2}{5})}^{-3/2}={\left(1-x\right)}^{\frac{-p}{q}}\\ \frac{8}{5}+S={\left(\frac{3}{5}\right)}^{-3/2}\\ \frac{8}{5}+S={\left(\frac{5}{3}\right)}^{3/2}\\ \Rightarrow S=\frac{5\sqrt{5}}{3\sqrt{3}}-\frac{8}{5}\end{array}$