Question

Evaluate $\underset{x\to 1}{lim}\frac{\left[\sum _{K=1}^{100}{x}^K\right]-100}{x-1}$

Answer 1

$\underset{x\to 1}{lim}\frac{\left[\sum _{K=1}^{100}{x}^K-100\right]}{x-1}$
$=\underset{x\to 1}{lim}\left[\frac{x+x^2+x^3+......+x^{100}-100}{x-1}\right]$
$=\underset{x\to 1}{lim}\left[\frac{1+2x+3x^2+......+100x^{99}}{1}\right]$ (Using L' Hospital's)
$=5050$.