Question

If x is so small show that √1-x/1+x=1-x+x2/2

Answer 1

$\sqrt{ \frac{1-x}{1+x} } \\ \\ = \sqrt{ \frac{1-x}{1+x} } \times \sqrt{ \frac{1-x}{1-x} }\\ \\ = \sqrt{ \frac{(1-x)^2}{1-x^2} }\\ \\ = \frac{1-x}{ \sqrt{1-x^2} } \\ \\ =(1-x) \times (1-x^2)^{- \frac{1}{2} }\\ \\ Using\ binomial\ approximation\ for\ small\ x,\ (1+x)^a =1+ax\\ \\ =(1-x) \times (1-(- \frac{1}{2})x^2)$

$=(1-x)(1+ \frac{x^2}{2})\\ \\ = 1+ \frac{x^2}{2}-x- \frac{x^3}{2} \\ \\ Ignoring\ x^3\ terms\ (since\ x\ is\ small)\\ \\ =1-x+ \frac{x^2}{2}$