Sum to infinity of GP

Assume \(a\ne0\). If \(a=0\), every term and every partial sum is zero. We now consider the infinite geometric series

\[a + ar + ar^2 + ar^3 + \dots\]

What does it even mean? The correct interpretation is that this infinite series, if it exists, is the limiting value of the sum \(S_n\) of the first \(n\) terms of the geometric sequence

\[a + ar + ar^2 + ar^3 + \dots + ar^{n-1}.\]

For \(r\ne1\), this sum can be written as \(\frac{a(1-r^n)}{1-r}\); we want to understand whether this expression converges to a finite value as \(n\) tends to infinity.

We consider the first scenario when \(|r| \ge 1\). For \(r=1\), \(S_n=na\) is unbounded; for \(r=-1\), the partial sums alternate between \(a\) and \(0\). For \(|r|>1\), the partial sums are unbounded in magnitude. In each case there is no finite limiting sum, so the series is said to diverge.

The second scenario is when \(|r| < 1\). In this second case, the magnitude \(|r|^n\) approaches \(0\), so \(r^n\) approaches \(0\), i.e., \(r^{n} \to 0\) as \( n \to \infty\). Thus, the sum \(S_n\) tends to the limiting value of
\[ S_\infty = \frac{a(1-0)}{1-r} = \frac{a}{1-r}. \]

In summary, we have that the following formula for sum to infinity (it exists):

When \(|r|<1\), as \(n \to \infty\), we have \(S_n \to S_\infty\), where
\[ S_\infty = \frac{a}{1-r}, \quad \qquad |r|<1. \]

In this case, we say that the infinite geometric series converges (to the value \(S_\infty\). Equivalently, we say that the infinite geometric series is convergent,