The limiting sum of a geometric series

ContentThe limiting sum of a geometric seriesWe have seen that the sum of the first \(n\) terms of a geometric series with first term \(a\) and common ratio \(r\) is\[S_n = \dfrac{a(1 - r^n)}{1 - r}, \quad\text{for } r\neq 1.\]In the case when \(r\) has magnitude less than 1, the term \(r^n\) approaches 0 as \(n\) becomes very large. So, in this case, the sequence of partial sums \(S_1,S_2,S_3,\dots\) has a limit:\[\lim_{n\to \infty} S_n = \lim_{n\to \infty} \dfrac{a(1 - r^n)}{1 - r} = \dfrac{a}{1 - r}.\]The value of this limit is called the limiting sum of the infinite geometric series. The values of the partial sums \(S_n\) of the series get as close as we like to the limiting sum, provided \(n\) is large enough.The limiting sum is usually referred to as the sum to infinity of the series and denoted by \(S_\infty\). Thus, for a geometric series with common ratio \(r\) such that \(|r|<1\), we have\[S_\infty = \lim_{n\to \infty} S_n = \dfrac{a}{1 - r}.\] Example Find the limiting sum f...