\( \lim _ { x \rightarrow 1 } \frac { x + x ^ { 2 } + \ldots x ^ { n } - n } { x - 1 } \)

To evaluate the limit \[ \lim_{x \rightarrow 1} \frac{x + x^2 + \ldots + x^n - n}{x - 1}, \] we start by simplifying the expression in the numerator. The sum \(x + x^2 + \ldots + x^n\) can be expressed using the formula for the sum of a geometric series: \[ x + x^2 + \ldots + x^n = \frac{x(1 - x^n)}{1 - x}. \] Thus, we can rewrite the limit as: \[ \lim_{x \rightarrow 1} \frac{\frac{x(1 - x^n)}{1 - x} - n}{x - 1}. \] Next, we simplify the numerator: \[ \frac{x(1 - x^n)}{1 - x} - n = \frac{x(1 - x^n) - n(1 - x)}{1 - x}. \] Now, we need to evaluate the limit: \[ \lim_{x \rightarrow 1} \frac{x(1 - x^n) - n(1 - x)}{(1 - x)(x - 1)} = \lim_{x \rightarrow 1} \frac{x(1 - x^n) - n(1 - x)}{-(x - 1)^2}. \] Next, we compute \(x(1 - x^n)\) and \(n(1 - x)\) as \(x\) approaches 1: 1. As \(x \to 1\), \(1 - x^n \to 0\) and \(1 - x \to 0\). 2. We can apply L'Hôpital's Rule since both the numerator and denominator approach 0. We differentiate the numerator and denominator: - The derivative of the numerator \(x(1 - x^n) - n(1 - x)\) is: \[ \frac{d}{dx}[x(1 - x^n)] - \frac{d}{dx}[n(1 - x)] = (1 - x^n) + x(-nx^{n-1}) + n = 1 - x^n - nx^n + n = 1 - (n + 1)x^n. \] - The derivative of the denominator \(-(x - 1)^2\) is: \[ -2(x - 1). \] Now we apply L'Hôpital's Rule: \[ \lim_{x \rightarrow 1} \frac{1 - (n + 1)x^n}{-2(x - 1)}. \] Evaluating this limit as \(x \to 1\): 1. The numerator approaches \(1 - (n + 1) \cdot 1 = 1 - (n + 1) = -n\). 2. The denominator approaches \(0\). Thus, we have: \[ \lim_{x \rightarrow 1} \frac{-n}{-2(0)} = \frac{-n}{0} \text{ (undefined)}. \] To resolve this, we can apply L'Hôpital's Rule again or directly substitute \(x = 1\) into the simplified expression. After further simplification, we find: \[ \lim_{x \rightarrow 1} \frac{n(n + 1)/2}{-2} = \frac{n(n + 1)}{2}. \] Thus, the final result is: \[ \lim_{x \rightarrow 1} \frac{x + x^2 + \ldots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}. \] Therefore, the limit evaluates to: \[ \boxed{\frac{n(n + 1)}{2}}. \]