
# 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

$|z| > |w|.$

8.1: Geometric Series is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.