Question

Show that $\underset{x\to 1}{lim}\frac{x+x^2+x^3+....+x^n-n}{x-1}$ is $\frac{n(n+1)}{2}$.

Answer 1

${lim}_{x\to 1}\frac{\frac{x^{n+1}-x}{x-1}-n}{x-1}$

${lim}_{x\to 1}\frac{x^{n+1}-x-nx+n}{(x-1{)}^2}$

Applying L's Hospital twice, we get

${lim}_{x\to 1}\frac{n(n+1)x^{n-1}}{2}$

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