# example of ratio test

Consider the sequence given by $a_{n}=x^{n}$ (geometric progression) where $|x|<1$. Then the series

 $\sum_{j=0}^{\infty}a_{n}$

converges.  To see this, we can use the ratio test. We need to consider the sequence $|a_{n+1}/a_{n}|$. But for any  $n\geq 0$  we have (when  $x\neq 0$)

 $\left|\frac{a_{n+1}}{a_{n}}\right|=\left|\frac{x^{n+1}}{x^{n}}\right|=|x|<1,$

and therefore the series converges. The ratio test and the previous argument shows that the geometric series diverges for  $|x|>1$.

Title example of ratio test ExampleOfRatioTest 2013-03-22 15:03:20 2013-03-22 15:03:20 drini (3) drini (3) 9 drini (3) Example msc 26A06 msc 40A05