Geometric Series

The geometric series is the sum of an infinite number of terms where two successive terms have a constant ratio. An example of a geometric series is:

\[1 + q + q^2 + q^3 + \dots = \sum_{i=0}^\infty q^i\]

In this series each term is the geometric mean of its previous and subsequent term, hence the name geometric series.

Proof: The geometric mean for two numbers \(a\) and \(b\) is defined as \(\sqrt{ab}\). It follows \(\sqrt{q^{i-1}q^{i+1}} = \sqrt{q^{i-1+i+1}} = \sqrt{q^{2i}} = q^i\)

The closed form of a geometric series for \(\vert q\vert < 1\) is

\[\sum_{i=0}^\infty q^i = \frac{1}{1-q}\]

Proof: If we compute the difference of the following finite series

\[\sum_{i=0}^n q^i - q\sum_{i=0}^n q^i\]

we get

\[\begin{split} \sum_{i=0}^n q^i - q\sum_{i=0}^n q^i &= (1+q+q^2+\dots+q^n) - (q+q^2+\dots+q^{n+1}) \\[2ex] &= 1-q^{n-1} \\[2ex] \left(1-q \right) \sum_{i=0}^n q^i &= 1-q^{n-1} \\[2ex] \sum_{i=0}^n q^i &= \frac{1-q^{n-1}}{1-q} \qquad \text{for } q\neq 1 \\[2ex] \end{split}\]

Now, if \(n\to \infty\) it follows for \(\vert q\vert < 1\) that \(q^{n-1}\to 0\) and we get

\[\sum_{i=0}^\infty q^i = \frac{1}{1-q}\]