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: $1, 2, 3, 4, 5, 6, 7, 8, 9, \dots$

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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!