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

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

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

Next, expand \((1-2x)^{-\frac{1}{2}}\):

\(1 + \frac{1}{2}(2x) + \frac{1}{2} \cdot \frac{3}{2} (2x)^2 = 1 + x + \frac{3}{2}x^2\).

Now, multiply the expansions:

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

Calculate the product up to \(x^2\):

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

Combine like terms:

\(1 + 2x + 2x^2\).