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Q. $\displaystyle \lim_{x \to 1}$$\frac{x+x^{2}+...+x^{n}-n}{x-1}$ is

Limits and Derivatives

Solution:

We have
$\displaystyle \lim_{x \to 1}$$\left\{\frac{x+x^{2}+...+x^{n}-n}{x-1}\right\}$
$=\displaystyle \lim_{x \to 1}$$\left\{\frac{x+x^{2}+...+x^{n}-1-1-1-...-1}{x-1}\right\}$
$=\displaystyle \lim_{x \to 1}$$\left\{\frac{\left(x-1\right)+\left(x^{2}-1\right)+\left(x^{3}-1\right)+...+\left(x^{n}-1\right)}{x-1}\right\}$
$=\displaystyle \lim_{x \to 1}$$\left\{1 + \left(x+ 1\right) + \left(x^{2} + x +1\right) + ...\right\}$
$=1+2+3+...+n$
$=\frac{n\left(n+1\right)}{2}$