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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).

Expand (1 - x)/(1 + x) in ascending powers of x up to and including the term in x^2. State the set of values of x for which the series expansion is valid. Find an approximation to the fourth root of 19/21, in the form p/q, where p and q are positive integers with no common factors.

Explore the binomial expansion of (1 - x)/(1 + x) up to x^2 and find the valid range of x. Also, learn to approximate the fourth root of 19/21 using a binomial series in this detailed high school-level problem.

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