$1+\frac{1}{{10}^2}+\frac{1.3}{1.2}.\frac{1}{{10}^4}+\frac{\mathrm{1.3.5}}{\mathrm{1.2.3}}.\frac{1}{{10}^6}+\mathrm{...}\infty =$

$1+\frac{1}{{10}^2}+\frac{1.3}{1.2}.\frac{1}{{10}^4}+\frac{\mathrm{1.3.5}}{\mathrm{1.2.3}}.\frac{1}{{10}^6}+\mathrm{...}\infty$

$1+\frac{1}{1!}\left(\frac{1}{{10}^2}\right)+\frac{1.3}{1.2}{\left(\frac{1}{{10}^2}\right)}^2+\mathrm{...}\infty$

$1+\frac{p}{1!}\left(\frac{x}{q}\right)+\frac{p(p+q)}{2!}{\left(\frac{x}{q}\right)}^2+\mathrm{...}\infty$

$={(1-x)}^{\frac{-p}{q}}$

$p=1,q=2,\frac{x}{q}=\frac{1}{100}\Rightarrow x=\frac{1}{50}$

$\Rightarrow {(1-\frac{1}{50})}^{-\frac{1}{2}}={\left(\frac{49}{50}\right)}^{-\frac{1}{2}}={\left(\frac{50}{49}\right)}^{\frac{1}{2}}$

$=\frac{5\sqrt{2}}{7}$