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.

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

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

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

Proof

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

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

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

$$S_n - rS_n = a - ar^{n + 1} \nonumber$$

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:

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

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

$$S = a \sum_{j = 0}^{\infty} r^j = \dfrac{a}{1 - r} \nonumber$$

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

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

Inside the integral we have the expression

$$\dfrac{1}{w - z} \nonumber$$

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

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

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

$$|z| < |w|. \nonumber$$

Similarly,

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

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

$$|z| > |w|. \nonumber$$