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.