Find the generating functions and the associated sequences of: (1-x)^-2

The expression \((1 - x)^{-2}\) represents a generating function. To find its associated sequence, we can expand \((1 - x)^{-2}\) as a power series.

### 1. Expansion of \((1 - x)^{-2}\)

We recognize that this function is of the form \((1 - x)^{-n}\), where \(n = 2\). For such a form, the binomial series expansion is given by:

\[
(1 - x)^{-n} = \sum_{k=0}^{\infty} \binom{n+k-1}{k} x^k
\]

For \(n = 2\), this becomes:

\[
(1 - x)^{-2} = \sum_{k=0}^{\infty} \binom{1+k}{k} x^k
\]

### 2. Finding the Coefficients

The binomial coefficient \(\binom{1+k}{k}\) simplifies as follows:

\[
\binom{1+k}{k} = \frac{(1+k)!}{k! \cdot 1!} = k+1
\]

Thus, the series expansion of \((1 - x)^{-2}\) is:

\[
(1 - x)^{-2} = \sum_{k=0}^{\infty} (k+1) x^k
\]

### 3. Associated Sequence

The coefficients of the power series represent the terms of the sequence associated with \((1 - x)^{-2}\). Therefore, the sequence associated with this generating function is:

\[
1, 2, 3, 4, 5, 6, 7, 8, 9, \dots
\]

This is the sequence of positive integers \(a_k = k + 1\), starting from \(a_0 = 1\).

### Summary

- **Generating Function**: \((1 - x)^{-2}\)
- **Associated Sequence**: \(1, 2, 3, 4, 5, 6, 7, 8, 9, \dots\)

Would you like further details or clarification?

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Here are five related questions:
1. What is the general formula for the binomial expansion of \((1 - x)^{-n}\)?
2. How can generating functions be applied to solve recurrence relations?
3. What other generating functions have sequences related to triangular numbers?
4. How do you derive a generating function for a geometric series?
5. What are some practical applications of generating functions in combinatorics?

**Tip**: Generating functions are powerful tools for encoding sequences and finding closed-form expressions for sums!

Explore the generating function (1-x)^-2 and its associated sequence through a detailed expansion using the binomial theorem. Understand how to derive the power series and identify the sequence of positive integers. Suitable for Grades 11-12 or early university students.

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