$\begin{align}  & \sqrt{\frac{1+x}{1-x}}=\sqrt{(1+x){{(1-x)}^{-1}}}={{(1+x)}^{\tfrac{1}{2}}}{{(1-x)}^{-\tfrac{1}{2}}} \\  & {{(1+x)}^{\tfrac{1}{2}}}=1+\frac{x}{2}+\frac{\tfrac{1}{2}(-\tfrac{1}{2})}{2}{{x}^{2}}+--- \\  & {{(1+x)}^{\tfrac{1}{2}}}=1+\frac{x}{2}+\frac{1}{8}{{x}^{2}}+---- \\  & \text{we are neglecting }{{x}^{3}}\text{ and higher terms of }x \\  & {{(1-x)}^{-\tfrac{1}{2}}}=1+\frac{x}{2}-\frac{(\tfrac{-1}{2})(-\tfrac{3}{2})}{2}{{x}^{2}}+--- \\  & {{(1-x)}^{-\tfrac{1}{2}}}=1+\frac{x}{2}+\frac{3{{x}^{3}}}{8}+--- \\  & \text{we are neglecting }{{x}^{3}}\text{ and higher terms of }x \\  & {{(1+x)}^{\tfrac{1}{2}}}{{(1-x)}^{-\tfrac{1}{2}}}\approx \left( 1+\frac{x}{2}+\frac{{{x}^{2}}}{8} \right)\left( 1+\frac{x}{2}+\frac{3{{x}^{2}}}{8} \right) \\  & \text{we are neglecting }{{x}^{3}}\text{ and higher terms of }x \\  & {{(1+x)}^{\tfrac{1}{2}}}{{(1-x)}^{-\tfrac{1}{2}}}\approx 1+\frac{x}{2}+\frac{3{{x}^{2}}}{8}+\frac{x}{2}+\frac{{{x}^{2}}}{4}+\frac{{{x}^{2}}}{8}+--- \\  & {{(1+x)}^{\tfrac{1}{2}}}{{(1-x)}^{-\tfrac{1}{2}}}\approx 1+x+\frac{1}{2}{{x}^{2}}+-- \\  & \sqrt{\frac{1+x}{1-x}}\approx 1+x+\frac{1}{2}{{x}^{2}}+-- \\  & \text{Putting }\frac{1}{7}\text{ into LHS and RHS} \\  & \sqrt{\frac{1+\tfrac{1}{7}}{1-\tfrac{1}{7}}}\approx 1+\frac{1}{7}+\frac{1}{2}{{\left( \frac{1}{7} \right)}^{2}}+--- \\  & \sqrt{\frac{\tfrac{8}{7}}{\tfrac{6}{7}}}=\sqrt{\frac{4}{3}}\approx 1+\frac{1}{7}+\frac{1}{98}+--- \\  & \frac{2}{\sqrt{3}}\approx \frac{98+14+1}{98} \\  & \sqrt{3}\approx \frac{196}{113} \\ \end{align}$