Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

8.1: Geometric Series

  • Page ID
    6516
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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.