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