# $\lim\limits_{x \uparrow 1} \exp (-\sum\limits_{n=0}^{\infty}x^n)$

I have to determine the following:

$\lim\limits_{x \uparrow 1} \exp (-\sum\limits_{n=0}^{\infty}x^n)$

My idea is the following:

$\lim\limits_{x \uparrow 1} \exp (-\sum\limits_{n=0}^{\infty}x^n) = \lim\limits_{x \uparrow 1} \exp (-\frac{1}{1-x^n})$=...

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You should have $\frac{1}{1-x}$ instead of $\frac{1}{1-x^n}$. Then, what is the conclusion of your idea? –  Antonio Vargas Dec 8 '13 at 23:32

By continuity of $\exp$ $$\lim_{x \uparrow 1} \exp \left(- \sum_{n=0}^\infty x^n \right)=\exp \left( \lim_{x \uparrow 1} - \sum_{n=0}^\infty x^n \right)=\exp \left( \lim_{x \uparrow 1} \frac{1}{x-1} \right)=\exp(-\infty)=0.$$