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 $n \ge 0$ 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 $|x|>1$ .