A calculus problem by KY Kong

$\begin{aligned} (1-x)^{-\frac{1}{2}} & = \small 1 + \frac{-\frac{1}{2}}{1}(-x) + \frac{-\frac{1}{2}\left( -\frac{3}{2}\right)}{1 \dot{} 2}(-x)^2 + \frac{-\frac{1}{2}\left( -\frac{3}{2}\right)\left( -\frac{5}{2}\right)}{1 \dot{} 2 \dot{}3 }(-x)^3 \\ & \small \quad + \frac{-\frac{1}{2}\left( -\frac{3}{2}\right)\left( -\frac{5}{2}\right)\left( -\frac{7}{2}\right)}{1 \dot{} 2 \dot{}3 \dot{} 4 }(-x)^4 + ... \\ & = 1 + \frac{x}{2} + \frac{3x^2}{8} + \frac{5x^3}{16} + \frac{35x^4}{128} + ... \\ \Rightarrow \frac{1}{\sqrt{1-\frac{1}{50}}} & = 1 + \frac{0.02}{2} + \frac{3\dot{} 0.02^2}{8} + \frac{5\dot{} 0.02^3}{16} + \frac{35\dot{} 0.02^4}{128} + ... \\ \sqrt{\frac{50}{49}} & = 1 + 0.01 + 0.00015 + 0.0000025 + 0.00000004375 \\ \frac{5\sqrt{2}}{7} & = 1.010152544 \\ \Rightarrow \sqrt{2} & = \frac{7}{5} \times 1.010152544 \\ & = \boxed{1.41421} \end{aligned}$