Question

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

(1) \( n \)

(2) \( \frac{n+1}{2} \)

(3) \( \frac{n(n+1)}{2} \)

(4) \( \frac{n(n-1)}{2} \)

Answer 1

Correct option is (3) \( \frac{n(n + 1)}2\)

\(\lim\limits_{x \to 1} \frac{x + x^2 + ....+ x^n -n}{x -1}\)     \(\left(\frac{n-n}{1-1} = \frac 00\ type\right)\)

\(= \lim\limits_{x \to 1} \frac {1 + 2x +...+ nx^{n -1}}1\)     (By D.L.H. Rule)

\(= 1 + 2 + 3 + ....+n\)

\(= \frac{n(n + 1)}2\)