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 $x\eq 0$ )

\\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 $|x|>1$ .