To expand \(\sqrt{\left( \frac{1-x}{1+x} \right)}\), we can write it as \((1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}\).

First, expand \((1-x)^{\frac{1}{2}}\) using the binomial series:

\((1-x)^{\frac{1}{2}} = 1 - \frac{1}{2}x + \frac{1}{8}x^2 + \cdots\)

Next, expand \((1+x)^{-\frac{1}{2}}\) using the binomial series:

\((1+x)^{-\frac{1}{2}} = 1 - \frac{1}{2}x + \frac{3}{8}x^2 + \cdots\)

Now, multiply these expansions together and collect terms up to \(x^2\):

\((1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}} = (1 - \frac{1}{2}x + \frac{1}{8}x^2)(1 - \frac{1}{2}x + \frac{3}{8}x^2)\)

\(= 1 - \frac{1}{2}x - \frac{1}{2}x + \frac{1}{4}x^2 + \frac{1}{8}x^2 - \frac{3}{16}x^2 + \cdots\)

\(= 1 - x + \frac{1}{2}x^2\)