the sum to infinity

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September 12th 2009, 07:55 AM
kasia.ogor

the sum to infinity

For the following geometric serie, find the range of values of x for which the sum to infinity of the serie exist.
$1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ...$
Please help me with that problem.
September 12th 2009, 10:17 AM
red_dog

Let $S_n=1+\frac{1}{x}+\frac{1}{x^2}+\ldots+\frac{1}{x^ {n-1}}$

Then $S_n=\frac{x}{1-x}\left[\left(\frac{1}{x}\right)^n-1\right]$

$S_n$ converges if $\left|\frac{1}{x}\right|<1\Rightarrow |x|>1\Rightarrow x\in(-\infty,-1)\cup(1,\infty)$

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