# Limit problem

• Apr 30th 2013, 05:02 AM
HK47
Limit 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 AM
Plato
Re: Limit problem
Quote:

Originally Posted by HK47
I´m having trouble with this limit problem, would appreciate help.
$\displaystyle lim_{x\to\1}\frac{x+x^2+...+x^n-n}{x-1}$

Since both numerator and denominator approach zero you can apply L'Hopital's Rule.

If you are not allowed to use that then the algebra becomes very messy.
Note for each positive integer $\displaystyle k$, $\displaystyle (x-1)$ is a factor of $\displaystyle x^k-1$.
• Apr 30th 2013, 08:21 AM
HK47
Re: 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 AM
Plato
Re: Limit problem
Quote:

Originally Posted by HK47
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*}
• May 1st 2013, 02:40 AM
HK47
Re: Limit problem
So the limit is infinity. Thank you! Didn´t think to factor the n of all things.
• May 1st 2013, 03:12 AM
Plato
Re: Limit problem
Quote:

Originally Posted by HK47
So the limit is infinity.

No! Actually the limit is $\displaystyle \frac{n(n+1)}{2}$.
• May 1st 2013, 03:26 AM
HK47
Re: Limit problem
Oh okay, right, n isnt infinity :)
• May 1st 2013, 04:18 AM
Plato
Re: Limit problem
Quote:

Originally Posted by HK47
Oh okay, right, n isnt infinity

n is never infinity, because infinity is not a number.