Let [tex]S_n[/tex] be the [tex]n[/tex]th partial sum of a geometric sequence with common ratio [tex]r[/tex] and first term [tex]a[/tex]. So the sequence is [tex]\{a,ar,ar^2,\cdots\}[/tex], and [tex]S_n[/tex] is
[tex]S_n=a+ar+ar^2+\cdots+ar^{n-2}+ar^{n-1}[/tex]
Multiplying both sides by [tex]r[/tex] gives
[tex]rS_n=ar+ar^2+ar^3+\cdots+ar^{n-1}+ar^n[/tex]
and subtracting [tex]rS_n[/tex] from [tex]S_n[/tex] gives
[tex]S_n-rS_n=a+(ar-ar)+(ar^2-ar^2)+\cdots+(ar^{n-1}-ar^{n-1})-ar^n[/tex]
[tex](1-r)S_n=a(1-r^n)[/tex]
[tex]S_n=a\dfrac{1-r^n}{1-r}[/tex]
If [tex]0<r<1[/tex], then [tex]r^n\to0[/tex] as [tex]n\to\infty[/tex] and so the sum approaches
[tex]\displaystyle\lim_{n\to\infty}S_n=\dfrac a{1-r}[/tex]
