Find the sum of the next series:
$a + ar+ ar^2+ \cdots + ar^n+ \cdots$
where $a>0$ and $r >0$.

Suppose that $r \geq 1$. The sequence $\{ a r^n \}$ is increasing , so $\lim a r^n$ does not exists, therefore this series does not converge (diverges).

Suppose $r< 1$ then
$s_n = a + ar+ ar^2+ \cdots + ar^{n-1}$
and
$rs_n = ar+ ar^2+ \cdots + ar^{n-1}+ar^{n}$
by subtraction,
$s_n = \dfrac{a(1-r^n)}{1-r}$
Since $\lim r^n= 0$ then $\lim s_n= a/1-r$. We have thus not only shown that the series converges but have found its sum.