If `n` is a positive integer, prove that `1-2n+(2n(2n-1))/(2!)-(2n(2n-1)(2n-2))/(3!)++(-1)^(n-1)(2n(2n-1)(n+2))/((n-1)!)=(-1)^(n+1)(2n)!//2(n !)^2dot`

Explanation

To prove the given statement, we will follow a systematic approach using the properties of binomial coefficients and the binomial theorem.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We need to prove that:
\[
1 - 2n + \frac{2n(2n-1)}{2!} - \frac{2n(2n-1)(2n-2)}{3!} + \ldots + (-1)^{n-1} \frac{2n(2n-1)(2n-2)\ldots(2n-n+1)}{(n-1)!} = (-1)^{n+1} \frac{(2n)!}{2(n!)^2}
...