Question

Evaluate the limit: $$ \lim_{x \to 1} \frac{x + x^2 + x^3 + \dots + x^n - n}{x - 1}. $$ 

• A. \( \frac{n(n+1)}{2} \)
• B. \( \frac{n+1}{2} \)
• C. \( \frac{2}{n} \)
• D. \( n \)

Hint:

For limits involving summations, use L'Hôpital’s Rule or recognize summation identities to simplify expressions.

Answer 1

The Correct Option is A

Step 1: Recognizing the sum The numerator is the sum of the first \( n \) powers of \( x \): \[ S = x + x^2 + x^3 + \dots + x^n - n. \] Rewriting using the formula for sum of a geometric series: \[ S = \frac{x(x^n - 1)}{x - 1} - n. \] Step 2: Applying L'Hôpital's Rule Differentiating numerator and denominator: \[ \frac{d}{dx} [x + x^2 + x^3 + \dots + x^n - n] = 1 + 2x + 3x^2 + \dots + nx^{n-1}. \] At \( x = 1 \), this simplifies to: \[ 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}. \] Step 3: Evaluating the limit \[ \lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n+1)}{2}. \]