# Geometric series

A geometric series is an infinite series in which each term is a fixed multiple of the one before it, written as \( a + ar + ar^{2} + ar^{3} + \cdots = \sum_{n=1}^{\infty} a r^{n-1} \), where \( a \) is the initial term and \( r \) is the ratio. It converges exactly when \( |r| < 1 \), and in that case its sum is \( a/(1-r) \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> The series is one of the few infinite sums whose value can be computed in closed form, and it serves as the benchmark against which many other series are tested.<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup><sup> • </sup><sup>[2](https://ocw.mit.edu/ans7870/18/18.013a/textbook/chapter30/section03.html)</sup>

| Key fact | Detail |
|---|---|
| Form | \( \sum_{n=1}^{\infty} a r^{n-1} = a + ar + ar^{2} + ar^{3} + \cdots \), with constant ratio \( r \) between successive terms<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> |
| Convergence | Converges if \( |r| < 1 \); diverges if \( |r| \ge 1 \)<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> |
| Sum | \( \dfrac{a}{1-r} \) for \( |r| < 1 \)<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> |
| Partial sums | \( S_k = \dfrac{a(1-r^{k})}{1-r} \) for \( r \ne 1 \)<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> |
| Complex form | \( \sum_{j=0}^{\infty} z^{j} = 1/(1-z) \) for complex \( |z| < 1 \); divergent for \( |z| \ge 1 \)<sup>[3](https://math.rice.edu/~semmes/math331.pdf)</sup> |
| As a power series | \( 1 + x + x^{2} + \cdots = \dfrac{1}{1-x} \) on \( (-1,1) \), the Taylor series of \( 1/(1-x) \) with radius of convergence 1<sup>[4](https://openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions)</sup> |
| Earliest recorded result | Euclid's *Elements* Book IX, Proposition 35, gives the finite geometric sum formula<sup>[5](https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX35.html)</sup> |

## Definition and sum

A geometric series is any series that can be written in the form \( a + ar + ar^{2} + ar^{3} + \cdots \). The number \( r \) is called the ratio because each term divided by the previous term equals \( r \), and \( a \) is the initial term. The example \( 1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots \) has \( a = 1 \) and \( r = 1/2 \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup> Series that appear in slightly different forms, with the index starting at a value other than 1 or with a shifted exponent, count as geometric as long as they can be rewritten in this form; for instance \( \sum_{n=0}^{\infty} (2/3)^{n+2} \) has initial term \( a = 4/9 \) and ratio \( r = 2/3 \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup>

**Partial sums.** The \( k \)-th partial sum is \( S_k = a + ar + \cdots + ar^{k-1} \). Multiplying by \( 1 - r \) cancels the interior terms and leaves \( (1-r)S_k = a(1 - r^{k}) \), so for \( r \ne 1 \),

\[ S_k = \frac{a(1 - r^{k})}{1 - r}. \]

When \( r = 1 \), every term equals \( a \) and \( S_k = ak \), which grows without bound for \( a > 0 \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup>

**Convergence criterion.** If \( |r| < 1 \), then \( r^{k} \to 0 \), so \( S_k \to a/(1-r) \) and the series converges to that value. If \( |r| \ge 1 \), the partial sums diverge. The criterion depends only on the magnitude of \( r \): a negative ratio makes the terms alternate in sign, but the convergence condition and the sum formula are unchanged apart from the sign carried by \( r \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup><sup> • </sup><sup>[6](https://learn.math101.ca/calculus-ii/geometric-series/)</sup> The same statement holds for complex ratios: \( \sum_{j=0}^{\infty} z^{j} \) converges absolutely to \( 1/(1-z) \) when \( |z| < 1 \) and does not converge when \( |z| \ge 1 \), since then \( |z^{j}| \ge 1 \) and the terms cannot tend to zero.<sup>[3](https://math.rice.edu/~semmes/math331.pdf)</sup>

## Historical development

The finite geometric sum is ancient. Proposition 35 of Book IX of Euclid's *Elements* states that for numbers in continued proportion \( a_1, a_2, \ldots, a_n, a_{n+1} \), the excess of the last over the first is to the first as the excess of the last over the second is to the sum of all the preceding terms; in modern notation this yields \( a + ar + \cdots + ar^{n-1} = a\,(r^{n}-1)/(r-1) \).<sup>[5](https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX35.html)</sup>

Archimedes of Syracuse used a geometric summation with ratio \( 1/4 \) in his *Quadrature of the Parabola*, where the areas added at successive steps of his approximation satisfy \( a_{i+1} = a_i/4 \), and his summation formula reads \( A_1 + \tfrac{1}{4}A_1 + \cdots + \tfrac{1}{4}A_n + \tfrac{1}{3}A_{n+1} = \tfrac{4}{3}A_1 \).<sup>[7](https://old.maa.org/press/periodicals/convergence/a-translation-of-evangelista-torricellis-quadratura-parabolae-archimedes-quadrature-of-the-parabola)</sup> This is a special case of the infinite geometric series solving the area of a parabolic segment problem.<sup>[8](https://journal.unnes.ac.id/journals/ujme/article/view/2136)</sup> The paradoxes of [Zeno of Elea](https://www.edgepedia.org/zeno-of-elea), formulated around 450 BC, raised the question of summing infinitely many quantities and are counted among the earliest evidence of engagement with infinite series.<sup>[9](https://iep.utm.edu/zenos-paradoxes/)</sup><sup> • </sup><sup>[10](https://www.math.purdue.edu/~eremenko/dvi/series1.pdf)</sup>

A history of infinite summation by Franka Miriam Brückler, a mathematician at the [University of Zagreb](https://www.edgepedia.org/university-of-zagreb), divides the subject's development into early ideas in Greek antiquity, first explicit considerations in the 14th and 15th centuries, gradual systematisation from the mid-17th to the mid-18th century, and final formalisation in the 1820s, with [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy) among the figures of the last phase.<sup>[11](https://hrcak.srce.hr/en/clanak/497361)</sup>

## Role in analysis

**Comparison and ratio tests.** Because the behavior of a geometric series is completely settled, it is the standard comparison series. The ratio test rests on the same idea: if the ratio of successive terms settles to a limit \( \rho \), the series eventually behaves like a geometric series with ratio \( \rho \), and the threshold \( \rho < 1 \) for absolute convergence carries over.<sup>[2](https://ocw.mit.edu/ans7870/18/18.013a/textbook/chapter30/section03.html)</sup><sup> • </sup><sup>[12](https://locusmath.org/books/openstax-calculus-2/5-6)</sup> For series with positive terms, the comparison tests likewise measure an unknown series against a geometric series or a p-series.<sup>[13](https://openstax.org/books/calculus-volume-2/pages/5-key-concepts)</sup>

**Power series.** The geometric series \( 1 + x + x^{2} + x^{3} + \cdots \) converges to \( 1/(1-x) \) on the interval \( (-1,1) \) and diverges outside it, so its radius of convergence is 1.<sup>[4](https://openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions)</sup> This identity is the starting point for representing many functions by power series: algebraic manipulation of \( 1/(1-x) \) yields expansions such as \( 1/(1+x^{3}) = 1 - x^{3} + x^{6} - x^{9} + \cdots \) for \( |x| < 1 \).<sup>[4](https://openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions)</sup> In complex analysis, \( \sum_{n=0}^{\infty} z^{n} \) is the [Taylor series](https://www.edgepedia.org/taylor-series) of \( f(z) = 1/(1-z) \) centered at 0.<sup>[14](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_16.html)</sup>

**Remainders.** For a convergent geometric series, the error made by truncating after \( n \) terms has an explicit formula, since both the partial sum and the full sum are known exactly; this is a distinguishing advantage over most convergent series.<sup>[15](https://ximera.osu.edu/mooculus/calculus2TextbookBySection/remaindersSection/remainders/digInRemaindersGeometricAndTelescoping)</sup>

## Applications

**Repeating decimals.** A repeating decimal is a geometric series in base 10. For example, \( 3.262626\ldots = 3 + \tfrac{26}{10^{2}} + \tfrac{26}{10^{4}} + \cdots \), a geometric series with \( a = 26/10^{2} \) and \( r = 1/10^{2} \), whose sum \( 26/99 \) gives \( 3.262626\ldots = 323/99 \).<sup>[1](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)</sup>

**Finance.** [Compound interest](https://www.edgepedia.org/compound-interest) follows a geometric pattern: each period the balance is multiplied by a fixed factor, so the growth of the account is described by a geometric series, which is the feature that gives compounding its power.<sup>[16](https://openstax.org/books/contemporary-mathematics/pages/6-4-compound-interest)</sup>

**Linear algebra and numerical methods.** The geometric series generalizes to matrices as the [Neumann series](https://www.edgepedia.org/neumann-series): if \( A \) is a square matrix with \( \|A\| < 1 \), then \( I - A \) is invertible and \( (I-A)^{-1} = \sum_{k} A^{k} \), proved by the same telescoping multiplication used for the scalar case.<sup>[17](https://www.math.csi.cuny.edu/~verzani/Classes/Old/F2015/MTH335/Notes/Chapter4/series-iteration.html)</sup> More generally, \( \sum_{n=0}^{\infty} A^{n} \) converges to \( (I-A)^{-1} \) whenever the spectral radius of \( A \) is below 1, a result used in the analysis of Markov chains and iterative methods.<sup>[18](https://repository.tudelft.nl/file/File_81a55a5e-4a7e-46cd-892e-7a608aa43b45?preview=1)</sup>

**Signal processing.** The geometric series identity \( \sum_{k=0}^{\infty} a r^{k} = a/(1-r) \) is used to evaluate the discrete-time [Fourier transform](https://www.edgepedia.org/fourier-transform) of geometric sequences; for \( x[n] = (1/2)^{n} u[n] \) it gives \( X(\Omega) = 1/(1 - (1/2)e^{-j\Omega}) \).<sup>[19](https://my.ece.utah.edu/~ece3500/Fall2016/notes/class22.html)</sup> [Z-transform](https://www.edgepedia.org/z-transform) computations rely on the same identity, for example writing the transform of the constant signal \( \{1, 1, 1, \ldots\} \) as \( 1 + z^{-1} + z^{-2} + \cdots = z/(z-1) \) for \( |z| > 1 \).<sup>[20](https://homepages.math.uic.edu/~jan/mcs472/mcs472notes/lec06.html)</sup>

## References

1. [Geometric Series, §5.2 Infinite Series, OpenStax Calculus Volume 2](https://openstax.org/books/calculus-volume-2/pages/5-2-infinite-series)
2. [Conditions for Absolute Convergence, 18.013A Calculus with Applications, MIT OpenCourseWare](https://ocw.mit.edu/ans7870/18/18.013a/textbook/chapter30/section03.html)
3. [Sequences and Series, Chapter 3 (geometric series for complex numbers), Rice University](https://math.rice.edu/~semmes/math331.pdf)
4. [Power Series and Functions, §6.1, OpenStax Calculus Volume 2](https://openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions)
5. [Euclid's Elements, Book IX, Proposition 35 (David Joyce, Clark University)](https://mathcs.clarku.edu/~djoyce/elements/bookIX/propIX35.html)
6. [Geometric Series, Calculus II guide, Math101.ca](https://learn.math101.ca/calculus-ii/geometric-series/)
7. [A Translation of Evangelista Torricelli's Quadratura Parabolae: Archimedes' Quadrature of the Parabola, Mathematical Association of America, Convergence](https://old.maa.org/press/periodicals/convergence/a-translation-of-evangelista-torricellis-quadratura-parabolae-archimedes-quadrature-of-the-parabola)
8. [Using the History of Circle and Parabolic Segment Areas as Learning Alternatives in Integral, Unnes Journal of Mathematics Education (2024)](https://journal.unnes.ac.id/journals/ujme/article/view/2136)
9. [Zeno's Paradoxes, Internet Encyclopedia of Philosophy](https://iep.utm.edu/zenos-paradoxes/)
10. [Infinite Series lecture notes, Purdue University](https://www.math.purdue.edu/~eremenko/dvi/series1.pdf)
11. [Franka Miriam Brückler, A Short History of Infinite Series, Osječki matematički list 25(2), 2025](https://hrcak.srce.hr/en/clanak/497361)
12. [Ratio and Root Tests study guide, Locus (OpenStax Calculus Volume 2 study guide)](https://locusmath.org/books/openstax-calculus-2/5-6)
13. [Key Concepts, Chapter 5, OpenStax Calculus Volume 2](https://openstax.org/books/calculus-volume-2/pages/5-key-concepts)
14. [Lecture 16: Taylor series, complex analysis notes, UC Berkeley](https://math.berkeley.edu/~avizeff/complex_analysis_S26/lecture_16.html)
15. [Remainders for Geometric and Telescoping Series, Ximera, Ohio State University](https://ximera.osu.edu/mooculus/calculus2TextbookBySection/remaindersSection/remainders/digInRemaindersGeometricAndTelescoping)
16. [Compound Interest, §6.4, OpenStax Contemporary Mathematics](https://openstax.org/books/contemporary-mathematics/pages/6-4-compound-interest)
17. [Neumann series and iterative methods, MTH 335 notes, College of Staten Island, CUNY](https://www.math.csi.cuny.edu/~verzani/Classes/Old/F2015/MTH335/Notes/Chapter4/series-iteration.html)
18. [Geometric series for matrices and the PageRank method, TU Delft repository](https://repository.tudelft.nl/file/File_81a55a5e-4a7e-46cd-892e-7a608aa43b45?preview=1)
19. [Geometric progression and the DTFT, ECE 3500 notes, University of Utah](https://my.ece.utah.edu/~ece3500/Fall2016/notes/class22.html)
20. [Z-Transforms and Linear Recursions, MCS 472 notes, University of Illinois Chicago](https://homepages.math.uic.edu/~jan/mcs472/mcs472notes/lec06.html)

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