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The limit expression you provided is:

\[
\lim_{{x \to 1}} \frac{x + x^2 + \cdots + x^n - n}{(2 - x)^n - 1}
\]

Let's break this down and evaluate the limit as \(x\) approaches 1.

### Step-by-Step Evaluation:
#### 1. **Simplify the Numerator:**
   The numerator is a sum of powers of \(x\), specifically:
   \[
   x + x^2 + \cdots + x^n - n
   \]
   This is a finite geometric series, excluding the subtraction of \(n\). The sum of the geometric series can be written as:
   \[
   S = x + x^2 + \cdots + x^n = \frac{x(1 - x^n)}{1 - x}
   \]
   Subtracting \(n\) gives:
   \[
   S - n = \frac{x(1 - x^n)}{1 - x} - n
   \]

#### 2. **Simplify the Denominator:**
   The denominator is:
   \[
   (2 - x)^n - 1
   \]
   At \(x = 1\), this expression becomes \((1)^n - 1 = 0\). So, we will need to use L'Hôpital's Rule because both the numerator and denominator tend to 0 as \(x \to 1\).

#### 3. **Apply L'Hôpital's Rule:**
   To resolve the indeterminate form, we differentiate the numerator and the denominator with respect to \(x\).

#### Differentiation:
   - For the numerator, differentiate the sum term by term.
   - For the denominator, apply the chain rule to differentiate \((2 - x)^n - 1\).

Let me compute this to confirm the limit evaluation.

Calculate the limit of the expression (x + x^2 + ... + x^n - n) / ((2 - x)^n - 1) as x approaches 1.

Explore the detailed calculation of the limit for the expression (x + x^2 + ... + x^n - n) / ((2 - x)^n - 1) as x approaches 1, utilizing concepts from calculus including geometric series and L'Hôpital's Rule. Suitable for college-level mathematics students.

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