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The problem asks to express the given function as a power series and find its domain and radius of convergence.

We are given the function:

\[
\frac{1}{(1 - x)^2}
\]

### Step 1: Recognize a known power series

The function \(\frac{1}{1 - x}\) can be expressed as the sum of a geometric series:

\[
\frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n
\]
for \(|x| < 1\).

### Step 2: Differentiate the geometric series

To find the power series for \(\frac{1}{(1 - x)^2}\), we differentiate the geometric series \(\frac{1}{1 - x}\) with respect to \(x\):

\[
\frac{d}{dx} \left( \frac{1}{1 - x} \right) = \frac{1}{(1 - x)^2}
\]

Differentiating the series term by term:

\[
\frac{d}{dx} \left( \sum_{n=0}^{\infty} x^n \right) = \sum_{n=0}^{\infty} \frac{d}{dx}(x^n) = \sum_{n=1}^{\infty} n x^{n-1}
\]

Thus, the power series for \(\frac{1}{(1 - x)^2}\) is:

\[
\frac{1}{(1 - x)^2} = \sum_{n=1}^{\infty} n x^{n-1}
\]

### Step 3: Radius and domain of convergence

The radius of convergence for this power series is the same as the original geometric series, which is \(|x| < 1\). Hence, the radius of convergence is:

\[
R = 1
\]

The domain of convergence is the open interval:

\[
(-1, 1)
\]

### Summary:

The power series representation is:

\[
\frac{1}{(1 - x)^2} = \sum_{n=1}^{\infty} n x^{n-1}
\]

The radius of convergence is \(R = 1\), and the domain of convergence is \((-1, 1)\).

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Would you like more details or have any questions?

Here are 5 related questions for practice:
1. How would you find the power series of \(\frac{1}{(1 - 2x)^2}\)?
2. What happens to the radius of convergence when the function changes to \(\frac{1}{(1 + x)^2}\)?
3. Can you derive the power series for \(\frac{1}{(1 - x)^3}\)?
4. What is the sum of the series \(\sum_{n=1}^{\infty} n (-x)^{n-1}\)?
5. How do you find the power series expansion of \(\frac{1}{(2 - x)^2}\)?

**Tip**: Differentiation and integration of known power series can simplify finding new power series for more complex functions.

Exprese las siguientes funciones como serie de potencias y encuentre su conjunto y radio de convergencia: \(\frac{1}{(1 - x)^2}\)

Learn how to express the function \(\frac{1}{(1 - x)^2}\) as a power series and determine its radius of convergence. This problem involves differentiating a geometric series and covers key concepts in calculus such as power series and radius of convergence. Suitable for advanced high school and undergraduate students.

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