I´m having trouble with this limit problem, would appreciate help.
$\displaystyle lim_{x\to\1}\frac{x+x^2+...+x^n-n}{x-1}$
Yeah, cant use the L´Hopital quite yet. My math textbook sure likes to stump me with these exercises, which can not be solved by provided means. At least I can post them here, so I dont have to rack my brains indefinitely.
Take note:
$\displaystyle \begin{align*} \frac{{x + {x^2} + \cdots + {x^n} - n}}{{x - 1}} &= \frac{{(x - 1) + ({x^2} - 1) + \cdots + ({x^n} - 1)}}{{x - 1}} \\&= \frac{{\sum\limits_{k = 1}^n {({x^k} - 1)} }}{{x - 1}}\\ &= \sum\limits_{k = 1}^n {\left( {\sum\limits_{j = 0}^{k - 1} {{x^j}} } \right)} \end{align*}$