1 Geometric Series
1.1 Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have constant ratio between the successive terms. In general, for coefficient \(a\) and common ratio \(r\) the general form of a geometric series is \[ a+ar+ar^2+\dots=\sum_{k=0}^\infty ar^k, \] and this representation is known as the generator form of the series.
For \(|r|<1\) the series converges and therefore has a closed form where the infinite sum can be written as \[ \sum_{k=0}^\infty ar^k= \frac{a}{1-r}. \] The sum of the first \(n\) terms of a geometric series, up to and including the \(r^{n-1}\) term, also has a closed form where \[ s_{n}=\sum_{k=0}^{n-1}ar^k=\begin{cases} a\left( \frac{{1-r^n}}{1-r} \right)&\text{if }r\neq 1 \\ an&\text{o.w.} \end{cases} \]