Consider the binomial expansion (1 − x)−2.
(i) [5 marks] Write down its first three terms in ascending powers
of x.
(ii) [5 marks] Find the coefficient of xn.
(iii) [5 marks] Find the range of values of x for which the expansion
is valid.
(iv) [5 marks] Hence, or otherwise, find the value of
∞Σ
n=1
n
2n .

### (i) First Three Terms in the Binomial Expansion of \((1 - x)^{-2}\)

The binomial expansion of \((1 - x)^{-k}\) is given by:
\[
(1 - x)^{-k} = \sum_{n=0}^{\infty} \binom{k+n-1}{n} x^n
\]
For \((1 - x)^{-2}\), we substitute \(k = 2\):
\[
(1 - x)^{-2} = \sum_{n=0}^{\infty} \binom{n+1}{n} x^n
\]
The binomial coefficients \(\binom{n+1}{n}\) simplify as follows:
\[
\binom{n+1}{n} = n+1
\]
Thus, the first three terms of the expansion in ascending powers of \(x\) are:
\[
(1 - x)^{-2} = 1 + 2x + 3x^2 + \cdots
\]

### (ii) Coefficient of \(x^n\)

From the general binomial expansion, the coefficient of \(x^n\) in the expansion of \((1 - x)^{-2}\) is given by:
\[
\binom{n+1}{n} = n+1
\]
Thus, the coefficient of \(x^n\) is \(n+1\).

### (iii) Range of Values of \(x\) for the Expansion to be Valid

The binomial expansion \((1 - x)^{-k}\) is valid for:
\[
|x| < 1
\]
Therefore, for \((1 - x)^{-2}\), the range of \(x\) is:
\[
|x| < 1
\]

### (iv) Finding the Value of \(\sum_{n=1}^{\infty} \frac{n}{2^n}\)

We are asked to find:
\[
\sum_{n=1}^{\infty} \frac{n}{2^n}
\]
This is a standard summation. To solve it, we can express it as a power series derivative.

Start with the geometric series:
\[
\sum_{n=0}^{\infty} x^n = \frac{1}{1 - x} \quad \text{for} \ |x| < 1
\]
Now differentiate both sides with respect to \(x\):
\[
\sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{(1 - x)^2}
\]
Multiply by \(x\) to adjust the powers:
\[
\sum_{n=1}^{\infty} n x^n = \frac{x}{(1 - x)^2}
\]
Now substitute \(x = \frac{1}{2}\) to find the value of the given series:
\[
\sum_{n=1}^{\infty} \frac{n}{2^n} = \frac{\frac{1}{2}}{\left(1 - \frac{1}{2}\right)^2} = \frac{\frac{1}{2}}{\left(\frac{1}{2}\right)^2} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2
\]

Thus, the value of the sum is:
\[
\sum_{n=1}^{\infty} \frac{n}{2^n} = 2
\]

---

Would you like further details on any part of this explanation or have any questions?

### Additional Questions to Explore:
1. How do we derive the general form of binomial expansion for negative exponents?
2. What is the binomial coefficient \(\binom{k+n-1}{n}\) in more detail?
3. How do we extend the expansion of \((1 - x)^{-2}\) beyond the first three terms?
4. Can you explore the radius of convergence for similar binomial expansions?
5. How can we apply these expansions to approximate functions?

**Tip:** When working with series and summations, geometric series and their derivatives are key tools for simplifying complex sums.

Consider the binomial expansion (1 − x)−2. (i) [5 marks] Write down its first three terms in ascending powers of x. (ii) [5 marks] Find the coefficient of xn. (iii) [5 marks] Find the range of values of x for which the expansion is valid. (iv) [5 marks] Hence, or otherwise, find the value of ∞Σ n=1 n 2n.

Explore the binomial expansion of (1 - x)^-2, its first three terms, the coefficient of x^n, and the range of values for convergence. Also, learn how to calculate the summation of ∞Σ n=1 n/2^n using series derivatives and geometric series. Suitable for advanced high school or undergraduate math students.

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