I´m having trouble with this limit problem, would appreciate help.

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

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- Apr 30th 2013, 05:02 AMHK47Limit problem
I´m having trouble with this limit problem, would appreciate help.

$\displaystyle lim_{x\to\1}\frac{x+x^2+...+x^n-n}{x-1}$ - Apr 30th 2013, 05:57 AMPlatoRe: Limit problem
- Apr 30th 2013, 08:21 AMHK47Re: Limit problem
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.

- Apr 30th 2013, 08:43 AMPlatoRe: Limit problem
Take note:

$\displaystyle \begin{align*} \frac{{x + {x^2} + \cdots + {x^n} - n}}{{x - 1}} &= \frac{{(x - 1) + ({x^2} - 1) + \cdots + ({x^n} - 1)}}{{x - 1}} \\&= \frac{{\sum\limits_{k = 1}^n {({x^k} - 1)} }}{{x - 1}}\\ &= \sum\limits_{k = 1}^n {\left( {\sum\limits_{j = 0}^{k - 1} {{x^j}} } \right)} \end{align*}$ - May 1st 2013, 02:40 AMHK47Re: Limit problem
So the limit is infinity. Thank you! Didn´t think to factor the n of all things.

- May 1st 2013, 03:12 AMPlatoRe: Limit problem
- May 1st 2013, 03:26 AMHK47Re: Limit problem
Oh okay, right, n isnt infinity :)

- May 1st 2013, 04:18 AMPlatoRe: Limit problem