8.1: Geometric Series

Having a detailed understanding of geometric series will enable us to use Cauchy’s integral formula to understand power series representations of analytic functions. We start with the definition:

Definition: Finite Geometric Series

A finite geometric series has one of the following (all equivalent) forms.

\[\begin{align} S_n &= a(1 + r + r^2 + r^3 + ... + r^n) \\[4pt] &= a + ar + ar^2 + ar^3 + ... + ar^n \\[4pt] &= \sum_{j = 0}^{n} ar^j \\[4pt] &= a \sum_{j = 0}^{n} r^j \end{align}\]

The number \(r\) is called the ratio of the geometric series because it is the ratio of consecutive terms of the series.

Theorem \(\PageIndex{1}\)

The sum of a finite geometric series is given by

\[S_n = a(1 + r + r^2 + r^3 + ... + r^n) = \dfrac{a (1 - r^{n + 1})}{1 - r}.\]

Proof

This is a standard trick that you’ve probably seen before.

\[\begin{array} {rclcc} {S_n} & = & {a + } & { ar + ar^2 + ... + ar^n} & {} \\ {rS_n} & = & { } & {ar + ar^2 + ... + ar^n} & {+ ar^{n + 1}} \end{array}\]

When we subtract these two equations most terms cancel and we get

\[S_n - rS_n = a - ar^{n + 1}\]

Some simple algebra now gives us the formula in Equation 8.2.2.

Definition: Infinite Geometric Series

An infinite geometric series has the same form as the finite geometric series except there is no last term:

\[S = a + ar + ar^2 + ... = a \sum_{j = 0}^{\infty} r^j.\]

Note

We will usually simply say ‘geometric series’ instead of ‘infinite geometric series’.

Theorem \(\PageIndex{2}\)

If \(|r| < 1\) then the infinite geometric series converges to

\[S = a \sum_{j = 0}^{\infty} r^j = \dfrac{a}{1 - r}\]

If \(|r| \ge 1\) then the series does not converge.

Proof

This is an easy consequence of the formula for the sum of a finite geometric series. Simply let \(n \to \infty\) in Equation 8.2.2.

Note

We have assumed a familiarity with convergence of infinite series. We will go over this in more detail in the appendix to this topic.

Connection to Cauchy’s Integral Formula

Cauchy’s integral formula says

\[f(z) = \dfrac{1}{2\pi i} \int_C \dfrac{f(w)}{w - z} \ dw.\]

Inside the integral we have the expression

\[\dfrac{1}{w - z}\]

which looks a lot like the sum of a geometric series. We will make frequent use of the following manipulations of this expression.

\[\dfrac{1}{w - z} = \dfrac{1}{w} \cdot \dfrac{1}{1 - z/w} = \dfrac{1}{w} (1 + (z/w) + (z/w)^2 + ...)\]

The geometric series in this equation has ratio \(z/w\). Therefore, the series converges, i.e. the formula is valid, whenever \(|z/w| < 1\), or equivalently when

\[|z| < |w|.\]

Similarly,

\[\dfrac{1}{w - z} = -\dfrac{1}{z} \cdot \dfrac{1}{1 - w/z} = - \dfrac{1}{z} (1 + (w/z) + (w/z)^2 + ...)\]

The series converges, i.e. the formula is valid, whenever \(|w/z| < 1\), or equivalently when

\[|z| > |w|.\]