To find the first three terms of the expansion of $\left(\frac{x-1}{x+1}\right)^n$, we can apply the binomial series expansion for $\left(1 + u\right)^n$, where $u = \frac{-2}{x+1}$.

Let's go through the steps:

1. Rewrite the expression:

   $\left(\frac{x-1}{x+1}\right)^n = \left(1 + \frac{-2}{x+1}\right)^n$

2. Apply the binomial expansion:

   The binomial expansion for $(1 + u)^n$ is:

   $(1 + u)^n = 1 + nu + \frac{n(n-1)}{2}u^2 + \cdots$

3. Substitute $u = \frac{-2}{x+1}$:

   Using this in the binomial expansion gives the first three terms:

   $\left(\frac{x-1}{x+1}\right)^n = 1 + n\left(\frac{-2}{x+1}\right) + \frac{n(n-1)}{2}\left(\frac{-2}{x+1}\right)^2 + \cdots$

4. Simplify the first three terms:

   [ \left(\frac{x-1}{x+1}\right)^n = 1 - \frac{2n}{x+1} + \frac