Let's work through the three parts of this problem.

(i) Expand $\left( \frac{1 - x}{1 + x} \right)^n$ in ascending powers of $x$ up to and including the term in $x^2$.

We begin by expressing $\left( \frac{1 - x}{1 + x} \right)^n$ as a binomial series. Start by simplifying:

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

Now, expand this in powers of $x$ up to $x^2$. The first few terms of the binomial expansion of $(1 - 2x + x^2)^n$ are:

$(1 - 2x + x^2)^n \approx 1 - 2nx + \frac{n(n-1)}{2}(4x^2) + \cdots$

So, the expansion up to the term in $x^2$ is:

$(1 - 2x + x^2)^n \approx 1 - 2nx + 2n(n-1)x^2.$

(ii) State the set of values of $x$ for which the series expansion is valid.

The expansion we used is a binomial series expansion. For a binomial series expansion of the form $(1 + ax)^n$ to be valid, we require:

$|ax| < 1.$

In our case, the expression we expanded was $(1 - 2x + x^2)^n$. To ensure convergence of this series, we must ensure that $|x| < 1$ for the binomial series expansion to hold.

Thus, the series expansion is valid for:

$|x| < 1.$

(iii) Find an approximation to the fourth root of $\frac{19}{21}$, in the form $\frac{p}{q}$, where $p$ and $q$ are positive integers with no common factors.

To approximate the fourth root of $\frac{19}{21}$, we need to expand $\left( \frac{19}{21} \right)^{1/4}$ using the binomial series.

First, express $\frac{19}{21}$ as:

$\frac{19}{21} = 1 - \frac{2}{21}.$

Now, we expand $(1 - \frac{2}{21})^{1/4}$ using the binomial expansion for $(1 + x)^n$, where $n = \frac{1}{4}$ and $x = -\frac{2}{21}$:

$(1 - \frac{2}{21})^{1/4} \approx 1 + \frac{1}{4} \left( -\frac{2}{21} \right) = 1 - \frac{1}{42}.$

Thus, the approximation is:

$\left( \frac{19}{21} \right)^{1/4} \approx 1 - \frac{1}{42} = \frac{41}{42}.$

Since $41$ and $42$ have no common factors, the required approximation is:

$\frac{p}{q} = \frac{41}{42}.$

Do you want any further details or have any questions about this? Here are some related questions:

1. How does the binomial expansion change when higher-order terms are included?
2. Can you apply the binomial series expansion to complex numbers?
3. What other methods can be used to approximate roots of fractions like $\frac{19}{21}$?
4. Why is the binomial expansion restricted to certain values of $x$?
5. How do you derive higher-order approximations for roots like $\frac{19}{21}$?

Tip: When expanding a binomial expression, always ensure the series converges by checking the domain of the variable involved (e.g., ensuring $|x| < 1$ in certain cases).