Geometric series

The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.

Another geometric series (coefficient a = 4/9 and common ratio r = 1/9) shown as areas of purple squares. The total purple area is S = a / (1 - r) = (4/9) / (1 - (1/9)) = 1/2, which can be confirmed by observing that the unit square is partitioned into an infinite number of L-shaped areas each with four purple squares and four yellow squares, which is half purple.

In mathematics, a geometric series is a series in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric sequence forms a geometric series. The name indicates that each term is the geometric mean of its two neighbouring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighbouring terms.

In general, a geometric series is written as $a+ar+ar^{2}+ar^{3}+...$, where $a$ is the coefficient of each term and $r$ is the common ratio between adjacent terms. For example, the series

${\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots$

is geometric because each successive term can be obtained by multiplying the previous term by $1/2$.

Truncated geometric series $a+ar+ar^{2}+ar^{3}+\dots +ar^{n}$ are called "finite geometric series" in certain branches of mathematics, especially in 19th century calculus and in probability and statistics and their applications.

The standard generator form expression for the infinite geometric series is

$a+ar+ar^{2}+ar^{3}+\dots =\sum _{k=0}^{\infty }ar^{k}$

and the generator form expression for the finite geometric series is

$a+ar+ar^{2}+ar^{3}+\dots +ar^{n}=\sum _{k=0}^{n}ar^{k}.$

Geometric series have been studied in mathematics from at least the time of Euclid in his work, Elements, which explored geometric proportions. Archimedes further advanced the study through his work on infinite sums, particularly in calculating areas and volumes of geometric shapes (for instance calculating the area inside a parabola) and the early development of calculus. They serve as prototypes for frequently used mathematical tools such as Taylor series, Fourier series, and matrix exponentials.

Geometric series have been applied to model a wide variety of natural phenomena, such as the expansion of the universe where the common ratio r is defined by Hubble's constant and the decay of radioactive carbon-14 atoms where the common ratio r is defined by the half-life of carbon-14.

Parameters

The geometric series $a+ar+ar^{2}+ar^{3}+\dots$ is an infinite series derived from a special type of sequence called a geometric progression, which is defined by just two parameters: the initial term $a$ and the common ratio $r$. Finite geometric series $a+ar+ar^{2}+ar^{3}+\dots +ar^{n}$ have a third parameter, the final term's index $n.$

The following table shows several geometric series with various initial terms and common ratios.

a	r	Example series
$3$	$1$	$3+3+3+3+\dots$
$3$	$-1$	$3-3+3-3+3-3+\dots$·
$4$	$10$	$4+40+400+4000+40000+\dots$
${\frac {1}{2}}$	${\frac {1}{2}}$	${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+\dots$
$1$	$-{\frac {1}{2}}$	$1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{8}}+{\frac {1}{16}}-{\frac {1}{32}}+\dots$
$7$	${\frac {1}{10}}$	$7+0.7+0.07+0.007+0.0007+\dots$
$1$	${\frac {2}{3}}$	$1+{\frac {2}{3}}+{\frac {4}{9}}+{\frac {8}{27}}+{\frac {16}{81}}+\dots$
$9$	${\frac {1}{3}}$	$9+3+1+{\frac {1}{3}}+{\frac {1}{9}}+\dots$

Initial term a

The geometric series $a+ar+ar^{2}+ar^{3}+\dots$ has the same coefficient $a$ in every term. The first term of a geometric series is equal to this coefficient and is the parameter $a$ of that geometric series, giving it its common interpretation: the "initial term."

In generator form, $\sum _{k=0}^{\infty }ar^{k},$ this term is technically written $ar^{0}$ instead of the bare $a$. This is equivalent because $r^{0}=1$ for any number $r.$

In contrast, general power series $a_{0}+a_{1}r+a_{2}r^{2}+a_{3}r^{3}+\dots$ have coefficients $a_{k}$ that can vary from term to term. In other words, the geometric series is a special case of the power series. Connections between power series and geometric series are discussed below in the section § Connections to power series.

Common ratio r

The parameter $r$ is called the common ratio because it is the ratio of any term with the previous term in the series.

$r={\frac {t_{k+1}}{t_{k}}}={\frac {ar^{k+1}}{ar^{k}}}$

where $t_{k}$ represents the $k$-th term of the geometric series.

The common ratio $r$ can be thought of as a multiplier used to calculate each next term in the series from the previous term.

Complex common ratio

Complex geometric series (coefficient a = 1 and common ratio r = 0.5 e^iω₀t) converging to trace a circle. In the animation, each term of the geometric series is drawn as a vector twice: once at the origin and again within the head-to-tail vector summation that converges to the circle. The circle intersects the real axis at 2 (= 1/(1-1/2) when ω₀t = 0) and at 2/3 (= 1/(1-(-1/2)) when ω₀t = π).

The common ratio $r$ can also be a complex number given by $\vert r\vert e^{i\theta }$, where $|r|$ is the magnitude of the number as a vector in the complex plane, $\theta$ is the angle or orientation of that vector, $e$ is Euler's number, and $i^{2}=-1$. In this case, the expanded form of the geometric series is

$a+a\vert r\vert e^{i\theta }+a\vert r\vert ^{2}e^{2s\theta }+a\vert r\vert ^{3}e^{i3\theta }+\dots$

An example of how this behaves for $\theta$ values that increase linearly over time with a constant angular frequency $\omega _{0}$, such that $\theta =\omega _{0}t,$ is shown in the adjacent video. For $\theta =\omega _{0}t,$ the geometric series becomes

$a+a\vert r\vert e^{i\omega _{0}t}+a\vert r\vert ^{2}e^{2s\omega _{0}t}+a\vert r\vert ^{3}e^{i3\omega _{0}t}+\dots$

where the first term is a vector of length $a$ that does not change orientation and all the following terms are vectors of proportional lengths rotating in the complex plane at integer multiples of the fundamental angular frequency $\omega _{0}$, also known as harmonics of $\omega _{0}$. As the video shows, these sums trace a circle. The period of rotation around the circle is $2\pi /\omega _{0}$.

Convergence

The convergence of the geometric series with r=1/2 and a=1/2

The convergence of the geometric series with r=1/2 and a=1

Close-up view of the cumulative sum of functions within the range -1 < r < -0.5 as the first 11 terms of the geometric series 1 + r + r² + r³ + ... are added. The geometric series 1 / (1 - r) is the red dashed line.

The convergence of the geometric series depends on the value of the common ratio $r$ alone:

• If $\vert r\vert <1$, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude) and the series converges to ${\frac {a}{1-r}}$
• If $\vert r\vert >1$, the terms of the series become larger and larger in magnitude and the sum of the terms also gets larger and larger in absolute value, so the series diverges.
• If $\vert r\vert =1$, the series does not converge. When $r=1$, all the terms of the series are the same and the series grows to infinity. When $r=-1$, the terms take two values alternately and therefore, the sequence of partial sums of the terms oscillates between two values. Consider, for example, Grandi's series: $1-1+1-1+1+...$. Partial sums of the terms oscillate between 1 and 0. Thus, the sequence of partial sums does not converge to any finite value.

When the series converges, the rate and pattern of convergence depend on the value of the common ratio $r$. The rate of convergence gets slower as $|r|$ approaches $1$. If $r<0$ and $|r|<1$, adjacent terms in the geometric series alternate between positive and negative and the partial sums of the terms oscillate above and below their eventual limit, whereas if $r>0$ and $|r|<1$ then terms all share the same sign and the partial sums of the terms approach their eventual limit monotonically. More on this is described below in the section § Rate of convergence.

Sum

As an alternative to the algebraic derivation of the geometric series closed form formula, there is also the following geometric derivation. (TOP) Represent the first n+1 terms of s/a as areas of overlapped similar triangles. For example, the area of the biggest overlapped (red) triangle is bh/2 = (2)(1)/2 = 1, which is the value of the first term of the geometric series. The area of the second biggest overlapped (green) triangle is bh/2 = (2r^1/2)(r^1/2)/2 = r, which is the value of the second term of the geometric series. Each progressively smaller triangle has its base and height scaled down by another factor of r^1/2, resulting in a sequence of triangle areas 1, r, r², r³, ... which is equal to the sequence of terms in the normalized geometric series. (MIDDLE) In the order from largest to smallest, remove each triangle's overlapped area, which is always a fraction r of its area, and scale the remaining 1−r of the triangle's non-overlapped area by 1/(1−r) so the area of the formerly overlapped triangle, now the area of a non-overlapped trapezoid, remains the same. (BOTTOM) Aggregate the resulting n+1 non-overlapped trapezoids into a single non-overlapped trapezoid and calculate its area. The area of that aggregated trapezoid represents the value of the partial series. That area is equal to the outermost triangle minus the empty triangle tip: s_n/a = (1−rⁿ⁺¹) / (1−r), which simplifies to s/a = 1/(1−r) when n approaches infinity and |r| < 1.

For convenience, the sum of the geometric series is often denoted by $S$ and its partial sums (the sums of the series going up to only the nth term) are often denoted $S_{n}.$ These can be represented as follows:

Form	Notation
Standard form	$S=a+ar+ar^{2}+ar^{3}+ar^{4}+\dots ={\frac {a}{1-r}}$
Partial sums	$S_{n}=a+ar+ar^{2}+\dots +ar^{n-1}={\frac {a(1-r^{n})}{1-r}}$

The sum of the first $n$ terms of a geometric series, up to and including the $r^{n-1}$ term,

$${\begin{aligned}S_{n}&=ar^{0}+ar^{1}+\cdots +ar^{n-1}\\&=\sum _{k=0}^{n-1}ar^{k}=\sum _{k=1}^{n}ar^{k-1}\end{aligned}}$$

is given by the closed form

$${\begin{aligned}S_{n}={\begin{cases}an&r=1\\a\left({\frac {1-r^{n}}{1-r}}\right)&{\text{otherwise}}\end{cases}}\end{aligned}}$$

where r is the common ratio. The case $r=1$ is just simple addition, a case of an arithmetic series. One can derive the closed-form formula for the partial sums $S_{n}$ in the case $r\neq 1$ by simplifying the many terms as follows: $${\begin{aligned}S_{n}&=ar^{0}+ar^{1}+\cdots +ar^{n-1},\\rS_{n}&=ar^{1}+ar^{2}+\cdots +ar^{n},\\S_{n}-rS_{n}&=ar^{0}-ar^{n},\\S_{n}\left(1-r\right)&=a\left(1-r^{n}\right),\\S_{n}&=a\left({\frac {1-r^{n}}{1-r}}\right),{\text{ for }}r\neq 1.\end{aligned}}$$

As $r$ approaches 1, polynomial division or L'Hospital's rule recovers the case $S_{n}=an$. As $n$ approaches infinity, the absolute value of r must be less than one for the series to converge, and when it does, the series converge absolutely. The sum of the infinite series then becomes

$${\begin{aligned}S&=a+ar+ar^{2}+ar^{3}+ar^{4}+\cdots \\&=\sum _{k=0}^{\infty }ar^{k}\\&={\frac {a}{1-r}},{\text{ for }}|r|<1.\end{aligned}}$$

The formula also holds for complex $r$, with the corresponding restriction that the absolute value of $r$ is strictly less than one.

The question of whether an infinite series converges is fundamentally a question about the distance between two values: given enough terms, does the value of the partial sum get arbitrarily close to a finite limit value that it is approaching? In the above derivation of the closed form of the geometric series, the interpretation of the distance between two values was the distance between their locations on the number line or on the complex plane.

That is the most common interpretation of the distance between two numbers. However, the p-adic metric, which has become a critical notion in modern number theory, offers a definition of distance such that the geometric series 1 + 2 + 4 + 8 + ... with $a=1$ and $r=2$ actually does converge to ${\frac {a}{1-r}}={\frac {1}{1-2}}=-1$ even though $r$ is outside the typical convergence range $|r|<1$.

Proof of convergence

Proof without words of the formula for the sum of a geometric series – if |r| < 1 and n → ∞, the rⁿ term vanishes, leaving S_∞ = ⁠a/1 − r⁠

We can prove that the geometric series converges using the sum formula for a geometric progression: $${\begin{aligned}1+r+r^{2}+r^{3}+\cdots \ &=\lim _{n\to \infty }\left(1+r+r^{2}+\cdots +r^{n}\right)\\&=\lim _{n\to \infty }{\frac {1-r^{n+1}}{1-r}}.\end{aligned}}$$ The second equality is true because if $|r|<1,$ then $r^{n+1}\to 0$ as $n\to \infty$ and $${\begin{aligned}\left(1+r+r^{2}+\cdots +r^{n}\right)(1-r)&=\left((1-r)+(r-r^{2})+(r^{2}-r^{3})+\dots +(r^{n}-r^{n+1})\right)\\&=1+(-r+r)+(-r^{2}+r^{2})+\dots +(-r^{n}+r^{n})-r^{n+1}\\&=1-r^{n+1}.\end{aligned}}$$

Alternatively, a geometric interpretation of the convergence is shown in the adjacent diagram. The area of the white triangle is the series remainder

$S-S_{n}={\frac {ar^{n}}{1-r}}$

Each additional term in the partial series reduces the area of that white triangle remainder by the area of the trapezoid representing the added term. The trapezoid areas (i.e., the values of the terms) get progressively thinner and shorter and closer to the origin. In the limit, as the number of trapezoids approaches infinity, the white triangle remainder vanishes as it is filled by trapezoids and therefore $s_{n}$ converges to $s$, provided $|r|<1$. In contrast, if $|r|>1$, the trapezoid areas representing the terms of the series instead get progressively wider and taller and farther from the origin, not converging to the origin and not converging as a series.

Rate of convergence

Converging alternating geometric series with common ratio r = -1/2 and coefficient a = 1. (TOP) Alternating positive and negative areas. (MIDDLE) Gaps caused by addition of adjacent areas. (BOTTOM) Gaps filled by broadening and decreasing the heights of the separated trapezoids.

For the geometric series, one convenient measure of the convergence rate is how much the previous series remainder decreases due to the last term of the partial series. Given that the last term is $ar^{n}$ and the previous partial sum remainder is $S-S_{n}={\frac {ar^{n}}{1-r}}$, this measure of the convergence rate of the geometric series is ${\frac {S_{n+1}-S_{n}}{S-S_{n}}}={\frac {ar^{n}}{\left({\frac {ar^{n}}{1-r}}\right)}}=1-r,0\leq r\leq 1$.

The adjacent diagram provides a geometric interpretation of a converging alternating geometric series, where the areas corresponding to the negative terms are shown below the x-axis. When each positive area is paired with its adjacent smaller negative area, the result is a series of non-overlapping trapezoids, separated by gaps.

To eliminate these gaps, we broaden each trapezoid so that it spans the rightmost $1-r^{2}$ of the original triangle area, instead of just the rightmost $1-|r|$. However, to ensure the areas of the trapezoids remain consistent during this transformation, scaling is necessary. The required scaling factor $\lambda$ can be derived from the equation:

$\lambda (1-r^{2})=(1-|r|)$

Simplifying this gives:

$\lambda ={\frac {1-|r|}{1-r^{2}}}={\frac {1+r}{(1+r)(1-r)}}={\frac {1}{1-r}}$

where $-1<r\leq 0.$ Notice that if $r<0$, this scaling factor decreases the amplitude of the trapezoids to fill the gaps. Conversely, when $r>0$, the same scaling factor $1/(1-r)$ increases the amplitude of the trapezoids to account for the loss of the overlapped areas.

After the gaps are removed, pairs of terms in the converging alternating geometric series form a new converging geometric series with a common ratio $r^{2}$, reflecting the pairing of terms. The coefficient $a=1/(1-r)$ compensates for the gap-filling, and the degree of the partial series, now denoted as $m$ instead of $n$, emphasizes that terms have been paired.

Interestingly, similar to the case when $r>0$, the convergence rate for $r<0$ is given by:

${\frac {ar^{2m}}{S-S_{m-1}}}=1-r^{2}$

This is identical to the convergence rate of a non-alternating geometric series if its terms were paired similarly. Therefore, the convergence rate is independent of $n$ or $m$, and perhaps more surprisingly, it does not depend on the sign of the common ratio.

One perspective that sheds light on the symmetry of the convergence rate around $r=0$ is that each added term of the partial series contributes a finite amount to the infinite sum when $r=1$, and similarly, each added term contributes a finite amount to the infinite slope when $r=-1$.

Animation, showing convergence of partial sums of geometric progression $\sum \limits _{k=0}^{n}q^{k}$ (red line) to its sum ${1 \over 1-q}$ (blue line) for $|q|<1$.

History

Zeno of Elea (c.495 – c.430 BC)

Main article: Zeno's Paradoxes

2,500 years ago, Greek mathematicians believed that an infinitely long list of positive numbers must sum to infinity. Therefore, Zeno of Elea created a paradox when he demonstrated that in order to walk from one place to another, one must first walk half the distance there, and then half of the remaining distance, and half of that remaining distance, and so on, covering infinitely many intervals before arriving. In doing so, he partitioned a fixed distance into an infinitely long list of halved remaining distances, each of which has length greater than zero. Zeno's paradox revealed to the Greeks that their assumption about an infinitely long list of positive numbers needing to add up to infinity was incorrect.

Euclid of Alexandria (c.300 BC)

Elements of Geometry, Book IX, Proposition 35. "If there is any multitude whatsoever of continually proportional numbers, and equal to the first is subtracted from the second and the last, then as the excess of the second to the first, so the excess of the last will be to all those before it."

A geometric interpretation for the same case of common ratio r>1. (TOP) Represent the terms of a geometric series as the areas of overlapped similar triangles. (MIDDLE) From the largest to the smallest triangle, remove the overlapped left area portion (1/r) from the non-overlapped right area portion (1-1/r = (r-1)/r) and scale that non-overlapped trapezoid by r/(r-1) so its area is the same as the area of the original overlapped triangle. (BOTTOM) Calculate the area of the aggregate trapezoid as the area of the large triangle less the area of the empty small triangle at the large triangle's left tip. The large triangle is the largest overlapped triangle scaled by r/(r-1). The empty small triangle started as a but that area was transformed into a non-overlapped scaled trapezoid leaving an empty left area portion (1/r). However, that empty triangle of area a/r must also be scaled by r/(r-1) so its slope matches the slope of all the non-overlapped scaled trapezoids. Therefore, S_n = area of large triangle - area of empty small triangle = arⁿ⁺¹/(r-1) - a/(r-1) = a(rⁿ⁺¹-1)/(r-1).

Euclid's Elements of Geometry Book IX, Proposition 35, proof (of the proposition in adjacent diagram's caption):

Let AA', BC, DD', EF be any multitude whatsoever of continuously proportional numbers, beginning from the least AA'. And let BG and FH, each equal to AA', have been subtracted from BC and EF. I say that as GC is to AA', so EH is to AA', BC, DD'.

For let FK be made equal to BC, and FL to DD'. And since FK is equal to BC, of which FH is equal to BG, the remainder HK is thus equal to the remainder GC. And since as EF is to DD', so DD' to BC, and BC to AA' [Prop. 7.13], and DD' equal to FL, and BC to FK, and AA' to FH, thus as EF is to FL, so LF to FK, and FK to FH. By separation, as EL to LF, so LK to FK, and KH to FH [Props. 7.11, 7.13]. And thus as one of the leading is to one of the following, so (the sum of) all of the leading to (the sum of) all of the following [Prop. 7.12]. Thus, as KH is to FH, so EL, LK, KH to LF, FK, HF. And KH equal to CG, and FH to AA', and LF, FK, HF to DD', BC, AA'. Thus, as CG is to AA', so EH to DD', BC, AA'. Thus, as the excess of the second is to the first, so is the excess of the last is to all those before it. The very thing it was required to show.

The terseness of Euclid's propositions and proofs may have been a necessity. As is, the Elements of Geometry is over 500 pages of propositions and proofs. Making copies of this popular textbook was labor intensive given that the printing press was not invented until 1440. And the book's popularity lasted a long time: as stated in the cited introduction to an English translation, Elements of Geometry "has the distinction of being the world's oldest continuously used mathematical textbook." So being very terse was being very practical. The proof of Proposition 35 in Book IX could have been even more compact if Euclid could have somehow avoided explicitly equating lengths of specific line segments from different terms in the series. For example, the contemporary notation for geometric series (i.e., a + ar + ar² + ar³ + ... + arⁿ) does not label specific portions of terms that are equal to each other.

Also in the cited introduction the editor comments,

Most of the theorems appearing in the Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematicians such as Pythagoras (and his school), Hippocrates of Chios, Theaetetus of Athens, and Eudoxus of Cnidos. However, Euclid is generally credited with arranging these theorems in a logical manner, so as to demonstrate (admittedly, not always with the rigour demanded by modern mathematics) that they necessarily follow from five simple axioms. Euclid is also credited with devising a number of particularly ingenious proofs of previously discovered theorems (e.g., Theorem 48 in Book 1).

To help translate the proposition and proof into a form that uses current notation, a couple modifications are in the diagram. First, the four horizontal line lengths representing the values of the first four terms of a geometric series are now labeled a, ar, ar², ar³ in the diagram's left margin. Second, new labels A' and D' are now on the first and third lines so that all the diagram's line segment names consistently specify the segment's starting point and ending point.

Here is a phrase by phrase interpretation of the proposition:

Proposition	in contemporary notation
"If there is any multitude whatsoever of continually proportional numbers"	Taking the first n+1 terms of a geometric series Sn = a + ar + ar2 + ar3 + ... + arn
"and equal to the first is subtracted from the second and the last"	and subtracting a from ar and arn
"then as the excess of the second to the first, so the excess of the last will be to all those before it."	then (ar-a) / a = (arn-a) / (a + ar + ar2 + ar3 + ... + arn-1) = (arn-a) / Sn-1, which can be rearranged to the more familiar form Sn-1 = a(rn-1) / (r-1).

Similarly, here is a sentence by sentence interpretation of the proof:

Proof	in contemporary notation
"Let AA', BC, DD', EF be any multitude whatsoever of continuously proportional numbers, beginning from the least AA'."	Consider the first n+1 terms of a geometric series Sn = a + ar + ar2 + ar3 + ... + arn for the case r>1 and n=3.
"And let BG and FH, each equal to AA', have been subtracted from BC and EF."	Subtract a from ar and ar3.
"I say that as GC is to AA', so EH is to AA', BC, DD'."	I say that (ar-a) / a = (ar3-a) / (a + ar + ar2).
"For let FK be made equal to BC, and FL to DD'."
"And since FK is equal to BC, of which FH is equal to BG, the remainder HK is thus equal to the remainder GC."
"And since as EF is to DD', so DD' to BC, and BC to AA' [Prop. 7.13], and DD' equal to FL, and BC to FK, and AA' to FH, thus as EF is to FL, so LF to FK, and FK to FH."
"By separation, as EL to LF, so LK to FK, and KH to FH [Props. 7.11, 7.13]."	By separation, (ar3-ar2) / ar2 = (ar2-ar) / ar = (ar-a) / a = r-1.
"And thus as one of the leading is to one of the following, so (the sum of) all of the leading to (the sum of) all of the following [Prop. 7.12]."	The sum of those numerators and the sum of those denominators form the same proportion: ((ar3-ar2) + (ar2-ar) + (ar-a)) / (ar2 + ar + a) = r-1.
"And thus as one of the leading is to one of the following, so (the sum of) all of the leading to (the sum of) all of the following [Prop. 7.12]."	And this sum of equal proportions can be extended beyond (ar3-ar2) / ar2 to include all the proportions up to (arn-arn-1) / arn-1.
"Thus, as KH is to FH, so EL, LK, KH to LF, FK, HF."
"And KH equal to CG, and FH to AA', and LF, FK, HF to DD', BC, AA'."
"Thus, as CG is to AA', so EH to DD', BC, AA'."
"Thus, as the excess of the second is to the first, so is the excess of the last is to all those before it."	Thus, (ar-a) / a = (ar3-a) / S2. Or more generally, (ar-a) / a = (arn-a) / Sn-1, which can be rearranged in the more common form Sn-1 = a(rn-1) / (r-1).
"The very thing it was required to show."	Q.E.D.

Archimedes of Syracuse (c.287 – c.212 BC)

Archimedes' dissection of a parabolic segment into infinitely many triangles

Main article: The Quadrature of the Parabola

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles.

Archimedes' Theorem states that the total area under the parabola is 4/3 of the area of the blue triangle.

Archimedes determined that each green triangle has 1/8 the area of the blue triangle, each yellow triangle has 1/8 the area of a green triangle, and so forth.

Assuming that the blue triangle has area 1, the total area is an infinite sum:

$1\,+\,2\left({\frac {1}{8}}\right)\,+\,4\left({\frac {1}{8}}\right)^{2}\,+\,8\left({\frac {1}{8}}\right)^{3}\,+\,\cdots .$

The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on. Simplifying the fractions gives

$1\,+\,{\frac {1}{4}}\,+\,{\frac {1}{16}}\,+\,{\frac {1}{64}}\,+\,\cdots .$

This is a geometric series with common ratio 1/4 and the fractional part is equal to

$\sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.$

The sum is

${\frac {1}{1-r}}\;=\;{\frac {1}{1-{\frac {1}{4}}}}\;=\;{\frac {4}{3}}.$

This computation uses the method of exhaustion, an early version of integration. Using calculus, the same area could be found by a definite integral.

Nicole Oresme (c.1323 – 1382)

A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2.

In addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the series

${\frac {1}{2}}+{\frac {2}{4}}+{\frac {3}{8}}+{\frac {4}{16}}+{\frac {5}{32}}+{\frac {6}{64}}+{\frac {7}{128}}+\dots =2.$

His diagram for his geometric proof, similar to the adjacent diagram, shows a two dimensional geometric series.

The first dimension is horizontal, in the bottom row, representing the geometric series with initial value $a={\frac {1}{2}}$ and common ratio $r={\frac {1}{2}}$

$${\begin{aligned}S&={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+\dots =\sum _{n=1}^{\infty }{\frac {1}{2}}\left({\frac {1}{2}}\right)^{n-1}\\&={\frac {\frac {1}{2}}{1-{\frac {1}{2}}}}=1\end{aligned}}$$

The second dimension is vertical, where the bottom row is a new coefficient $a_{T}=S$ and each subsequent row above it is scaled by the same common ratio $r={\frac {1}{2}}$, making another geometric series

$${\begin{aligned}T&=a_{T}\left(1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\dots \right)\\&={\frac {a_{T}}{1-r}}={\frac {S}{1-r}}\\&={\frac {1}{1-{\frac {1}{2}}}}=2.\end{aligned}}$$

Although difficult to visualize beyond three dimensions, Oresme's insight generalizes to any dimension $d$. Using the sum of the $(d-1)$ dimensions of the geometric series as the coefficient $a$ in the $d$ dimension of the geometric series results in a d-dimensional geometric series converging to

${\frac {S^{d}}{a}}={\frac {1}{(1-r)^{d}}}$

within the range $\vert r\vert <1$.

Pascal's triangle exhibits the coefficients of these multi-dimensional geometric series,

${\begin{matrix}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\end{matrix}}$

$d$	$S^{d}$ (closed form)	$S^{d}$ (expanded form)
$1$	$1/(1-r)$	$1+r+r^{2}+r^{3}+r^{4}+\cdots$
$2$	$1/(1-r)^{2}$	$1+2r+3r^{2}+4r^{3}+5r^{4}+\cdots$
$3$	$1/(1-r)^{3}$	$1+3r+6r^{2}+10r^{3}+15r^{4}+\cdots$
$4$	$1/(1-r)^{4}$	$1+4r+10r^{2}+20r^{3}+35r^{4}+\cdots$
$d$	$1/(1-r)^{d}$	$1+dr+{d+1 \choose {2}}r^{2}+{d+2 \choose {3}}r^{3}+{d+3 \choose {4}}r^{4}+\cdots$

where, as usual, the series converge to these closed forms only when $\vert r\vert <1$.

Examples

• Grandi's series – Infinite series summing alternating 1 and -1 terms: 1 − 1 + 1 − 1 + ⋯
• 1 + 2 + 4 + 8 + ⋯ – Infinite series
• 1 − 2 + 4 − 8 + ⋯ – infinite seriesPages displaying wikidata descriptions as a fallback
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯ – Mathematical infinite series
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯ – mathematical infinite seriesPages displaying wikidata descriptions as a fallback
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯ – Infinite series equal to 1/3 at its limit
• A geometric series is a unit series (the series sum converges to one) if and only if |r| < 1 and a + r = 1 (equivalent to the more familiar form S = a / (1 - r) = 1 when |r| < 1). Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7).
• The terms of a geometric series are also the terms of a generalized Fibonacci sequence (F_n = F_n-1 + F_n-2 but without requiring F₀ = 0 and F₁ = 1) when a geometric series common ratio r satisfies the constraint 1 + r = r², which according to the quadratic formula is when the common ratio r equals the golden ratio (i.e., common ratio r = (1 ± √5)/2).
• The only geometric series that is a unit series and also has terms of a generalized Fibonacci sequence has the golden ratio as its coefficient a and the conjugate golden ratio as its common ratio r (i.e., a = (1 + √5)/2 and r = (1 - √5)/2). It is a unit series because a + r = 1 and |r| < 1, it is a generalized Fibonacci sequence because 1 + r = r², and it is an alternating series because r < 0.

A compact illustration of different perspectives on the geometric series to make perspective switching easier. Quadrant A1 contains the why perspective briefly describing why geometric series matter. Quadrant A2 contains the how perspective listing a snippet of Julia code that computes the geometric series. Quadrant A3 contains the what (algebra) perspective with the algebraic proof of the geometric series closed form formula. Quadrant A4 contains the what (geometry) perspective showing three steps of the geometric proof of the geometric series closed form formula. And the right margin shows the timeline of some of the historical insights about the geometric series.

Repeating decimals and binaries

Main article: Repeating decimal

A subset of all converging geometric series converge to decimal numbers that have repeated patterns that continue forever, for instance $0.7777\ldots _{10},$ $0.9999\ldots _{10},$ or $0.123412341234\ldots _{10}.$ Although fractions with infinitely repeated decimal patterns can only be approximated with finite-length decimal numbers, they can always be defined exactly as the ratio of two integers, and those two integers can be calculated using the geometric series formula. For example, the repeated decimal fraction $0.7777\ldots _{10}$ can be written as the geometric series

$0.7777\ldots _{10}\;=\;{\frac {7_{10}}{10_{10}}}\,+\,{\frac {7_{10}}{10_{10}}}{\frac {1}{10_{10}}}\,+\,{\frac {7_{10}}{10_{10}}}{\frac {1}{10_{10}^{2}}}\,+\,{\frac {7_{10}}{10_{10}}}{\frac {1}{10_{10}^{3}}}\,+\,\cdots$

where the coefficient expressed as a base ten fraction $a=7_{10}/10_{10}$ and common ratio $r=1/10_{10}$. The geometric series closed form reveals the two integers that specify the repeated pattern:

$0.7777\ldots _{10}\;=\;{\frac {a}{1-r}}\;=\;{\frac {7_{10}/10_{10}}{1-1/10_{10}}}\;=\;{\frac {7_{10}/10_{10}}{9_{10}/10_{10}}}\;=\;{\frac {7_{10}}{9_{10}}}.$

This approach extends beyond base-ten numbers. In fact, any fraction that has an infinitely repeated pattern in base-ten numbers also has an infinitely repeated pattern in numbers written in any other base. For example, looking at the binary representation of the number $0.7777\ldots _{10}$ reveals the binary fraction $0.110001110001110001\ldots _{2}$ where the binary pattern 110001 repeats indefinitely. This binary pattern can be written as a series of binary terms as

$0.110001110001110001\ldots _{2}\;=\;{\frac {110001_{2}}{1000000_{2}}}\,+\,{\frac {110001_{2}}{1000000_{2}}}{\frac {1}{1000000_{2}}}\,+\,{\frac {110001_{2}}{1000000_{2}}}{\frac {1}{1000000_{2}^{2}}}\,+\,{\frac {110001_{2}}{1000000_{2}}}{\frac {1}{1000000_{2}^{3}}}\,+\,\cdots$

where coefficient $a=110001_{2}/1000000_{2}$ expressed in base two $=49_{10}/64_{10}$ in base ten and the common ratio $r=1/1000000_{2}$ in base two $=1/64_{10}$ in base ten. Using the geometric series closed form as before,

$0.7777\ldots _{10}\;=\;0.110001110001110001\ldots _{2}\;=\;{\frac {a}{1-r}}\;=\;{\frac {49_{10}/64_{10}}{1-1/64_{10}}}\;=\;{\frac {49_{10}/64_{10}}{63_{10}/64_{10}}}\;=\;{\frac {49_{10}}{63_{10}}}\;=\;{\frac {7_{10}}{9_{10}}}.$

As an example that has four digits in the repeating decimal pattern, $0.123412341234..._{10}$ can be written as the geometric series

$0.123412341234\ldots _{10}\;=\;{\frac {1234_{10}}{10000_{10}}}\,+\,{\frac {1234_{10}}{10000_{10}}}{\frac {1}{10000_{10}}}\,+\,{\frac {1234_{10}}{10000_{10}}}{\frac {1}{10000_{10}^{2}}}\,+\,{\frac {1234_{10}}{10000_{10}}}{\frac {1}{10000_{10}^{3}}}\,+\,\cdots$

where coefficient $a=1234_{10}/10000_{10}$ and common ratio $r=1/10000_{10}.$ The geometric series closed form reveals the two integers that specify the repeated pattern:

$0.123412341234\ldots _{10}\;=\;{\frac {a}{1-r}}\;=\;{\frac {1234_{10}/10000_{10}}{1-1/10000_{10}}}\;=\;{\frac {1234_{10}/10000_{10}}{9999_{10}/10000_{10}}}\;=\;{\frac {1234_{10}}{9999_{10}}}.$

Connections to power series

Like the geometric series, a power series $a_{0}+a_{1}x+a_{2}x^{2}+\ldots$ has one parameter for a common variable raised to successive powers, denoted $x$ here, corresponding to the geometric series's r, but it has additional parameters $a_{0},a_{1},a_{2},\ldots ,$ one for each term in the series, for the distinct coefficients of each $x^{0},x^{1},x^{2},\ldots$, rather than just a single additional parameter $a$ for all terms, the common coefficient of $r^{n}$ in each term of a geometric series.

The geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies $a_{n}=a$ for all $n$ and $x=r$. This special class of power series plays an important role in mathematics, for instance for the study of ordinary generating functions in combinatorics and the summation of divergent series in analysis. Many other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.

Derivations of other power series formulas

Infinite series formulas

One can use simple variable substitutions to calculate some useful closed form infinite series formulas. For an infinite series containing only even powers of $r$, for instance, $$\sum _{k=0}^{\infty }ar^{2k}=\sum _{k=0}^{\infty }a(r^{2})^{k}={\frac {a}{1-r^{2}}}$$ and for odd powers only, $$\sum _{k=0}^{\infty }ar^{2k+1}=\sum _{k=0}^{\infty }(ar)(r^{2})^{k}={\frac {ar}{1-r^{2}}}$$

In cases where the sum does not start at k = 0, one can use a shift of the index of summation together with a variable substitution, $$\sum _{k=m}^{\infty }ar^{k}=\sum _{k=0}^{\infty }(ar^{m})r^{k}={\frac {ar^{m}}{1-r}}$$ The formulas given above are strictly valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if |r|_p < 1.

One can also differentiate to calculate formulas for related sums. For example, $$\sum _{k=1}^{\infty }kr^{k-1}={\frac {d}{dr}}\sum _{k=0}^{\infty }r^{k}={\frac {1}{(1-r)^{2}}}$$

This formula is only strictly valid for |r| < 1 as well. From similar derivations, it follows that, for |r| < 1, $$\sum _{k=0}^{\infty }kr^{k}={\frac {r}{\left(1-r\right)^{2}}}\,;\,\sum _{k=0}^{\infty }k^{2}r^{k}={\frac {r\left(1+r\right)}{\left(1-r\right)^{3}}}\,;\,\sum _{k=0}^{\infty }k^{3}r^{k}={\frac {r\left(1+4r+r^{2}\right)}{\left(1-r\right)^{4}}}$$

It is also possible to use complex geometric series to calculate the sums of some trigonometric series using complex exponentials and Euler's formula. For example, consider the proposition $$\sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {r\sin(x)}{1+r^{2}-2r\cos(x)}}$$

This can be proven via the fact that $$\sin(kx)={\frac {e^{ikx}-e^{-ikx}}{2i}}.$$Substituting this into the original series gives $$\sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {1}{2i}}\left[\sum _{k=0}^{\infty }\left({\frac {e^{ix}}{r}}\right)^{k}-\sum _{k=0}^{\infty }\left({\frac {e^{-ix}}{r}}\right)^{k}\right].$$

This is the difference of two geometric series with common ratios $e^{ix}/r$ and $e^{-ix}/r$, and so the proof of the original proposition follows via two straightforward applications of the formula for infinite geometric series and then rearrangement of the result using$${\frac {e^{ix}-e^{-ix}}{2i}}=\sin(x)$$ and $${\frac {e^{ix}+e^{-ix}}{2}}=\cos(x)$$ to complete the proof.

Finite series formulas

Like for the infinite series, one can use variable substitutions and changes of the index of summation to derive other finite power series formulas from the finite geometric series formulas. If one were to begin the sum not from k=1 or 0 but from a different value, say ⁠$m$⁠, then

$${\begin{aligned}\sum _{k=m}^{n}ar^{k-1}&={\begin{cases}{\frac {a(r^{m-1}-r^{n})}{1-r}}&{\text{if }}r\neq 1\\a(n-m+1)&{\text{if }}r=1\end{cases}}\\\sum _{k=m}^{n}ar^{k}&={\begin{cases}a(n-m+1)&{\text{if }}r=1\\{\frac {a(r^{m}-r^{n+1})}{1-r}}&{\text{if }}r\neq 1\end{cases}}\end{aligned}}$$

For a geometric series containing only even powers of $r$, either multiply the finite sum by $1-r^{2}$ to derive a closed form for the partial sums:$${\begin{aligned}(1-r^{2})\sum _{k=0}^{n}ar^{2k}&=a-ar^{2n+2}\\\sum _{k=0}^{n}ar^{2k}&={\frac {a(1-r^{2n+2})}{1-r^{2}}}\end{aligned}}$$

or, equivalently, take $r^{2}$ as the common ratio $r$ and use the standard formula.

For a series with only odd powers of ⁠$r$⁠, the same derivation via multiplying by ⁠$1-r^{2}$⁠ applies: $${\begin{aligned}(1-r^{2})\sum _{k=0}^{n}ar^{2k+1}&=ar-ar^{2n+3}\\\sum _{k=0}^{n}ar^{2k+1}&={\frac {ar(1-r^{2n+2})}{1-r^{2}}}&={\frac {a(r-r^{2n+3})}{1-r^{2}}}\end{aligned}}$$or, equivalently, take $ar$ for $a$ and $r^{2}$ for $r$ in the standard form.

Differentiating such formulas with respect to ⁠$r$⁠ can give the formulas

$$G_{s}(n,r):=\sum _{k=0}^{n}k^{s}r^{k}.$$

For example:

$${\frac {d}{dr}}\sum _{k=0}^{n}r^{k}=\sum _{k=1}^{n}kr^{k-1}={\frac {1-r^{n+1}}{(1-r)^{2}}}-{\frac {(n+1)r^{n}}{1-r}}.$$

An exact formula for any of the generalized sums $G_{s}(n,r)$ when $s\in \mathbb {N}$ is

$$G_{s}(n,r)=\sum _{j=0}^{s}\left\lbrace {s \atop j}\right\rbrace x^{j}{\frac {d^{j}}{dx^{j}}}\left[{\frac {1-x^{n+1}}{1-x}}\right]$$

where $\left\lbrace {s \atop j}\right\rbrace$ denotes a Stirling number of the second kind.

Applications

Economics

Main article: Time value of money

In economics, geometric series are used to represent the present value of an annuity (a sum of money to be paid in regular intervals).

For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end of the year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannot invest the money until one receives it. In particular, the present value of $100 one year in the future is $100 / (1 + $I$ ), where $I$ is the yearly interest rate.

Similarly, a payment of $100 two years in the future has a present value of $100 / (1 + $I$)² (squared because two years' worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100 per year in perpetuity is

$\sum _{n=1}^{\infty }{\frac {\$100}{(1+I)^{n}}},$

which is the infinite series:

${\frac {\$100}{(1+I)}}\,+\,{\frac {\$100}{(1+I)^{2}}}\,+\,{\frac {\$100}{(1+I)^{3}}}\,+\,{\frac {\$100}{(1+I)^{4}}}\,+\,\cdots .$

This is a geometric series with common ratio 1 / (1 + $I$ ). The sum is the first term divided by (one minus the common ratio):

${\frac {\$100/(1+I)}{1-1/(1+I)}}\;=\;{\frac {\$100}{I}}.$

For example, if the yearly interest rate is 10% ($I$ = 0.10), then the entire annuity has a present value of $100 / 0.10 = $1000.

This sort of calculation is used to compute the APR of a loan (such as a mortgage loan). It can also be used to estimate the present value of expected stock dividends, or the terminal value of a financial asset assuming a stable growth rate.

Computer science

• Algorithm analysis:
  • Geometric series are used to analyze the time complexity of recursive algorithms (like divide-and-conquer) and in amortized analysis for operations with varying costs, such as dynamic array resizing.
• Data structures:
  • Geometric series help in analyzing the space and time complexities of operations in data structures like balanced binary search trees and heaps.
• Computer graphics:
  • Geometric series are crucial in rendering algorithms for anti-aliasing, for mipmapping, and for generating fractals, where the scale of detail varies geometrically.
• Networking and communication:
  • Geometric series model retransmission delays in exponential backoff algorithms and are used in data compression and error-correcting codes for efficient communication.
• Probabilistic and randomized algorithms:
  • Geometric series are used in analyzing random walks, Markov chains, and geometric distributions, which are essential in probabilistic and randomized algorithms.

Fractal geometry

The interior of the Koch snowflake is a union of infinitely many triangles. In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.

If |r| < 1, representing each geometric series term as the area of a similar triangle creates a fractal in which more zooming in always reveals more similar triangles.

The area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is

$1\,+\,3\left({\frac {1}{9}}\right)\,+\,12\left({\frac {1}{9}}\right)^{2}\,+\,48\left({\frac {1}{9}}\right)^{3}\,+\,\cdots .$

The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is

$1\,+\,{\frac {a}{1-r}}\;=\;1\,+\,{\frac {\frac {1}{3}}{1-{\frac {4}{9}}}}\;=\;{\frac {8}{5}}.$

Thus the Koch snowflake has 8/5 of the area of the base triangle.

Trigonometric power series

We can find the power series expansion of the arctangent function using some knowledge of differentiation, integration and the sum of a geometric series.

The derivative of $f(x)=\arctan(u(x))$ is known to be $f'(x)={\frac {u'(x)}{(1+[u(x)]^{2})}}$. This is a standard result from integration derived as follows. Let $y$ and $u$ represent $f(x)$ and $u(x)$,

${\begin{aligned}y&=\arctan(u)\\\implies u&=\tan(y)&&\quad {\text{ in the range }}-\pi /2<y<\pi /2{\text{ and }}\\\implies u'&=\sec ^{2}y\cdot y'&&\quad {\text{ by applying the quotient rule to }}\tan(y)=\sin(y)/\cos(y),\\\implies y'&={\frac {u'}{\sec ^{2}y}}&&\quad {\text{ by dividing both sides by }}\sec ^{2}y,\\&={\frac {u'}{(1+\tan ^{2}y)}}&&\quad {\text{ by using the trigonometric identity derived by dividing }}\sin ^{2}y+\cos ^{2}y=1{\text{ by }}\cos ^{2}y,\\&={\frac {u'}{(1+u^{2})}}\end{aligned}}$

Therefore, letting the arctan function is the integral

${\begin{aligned}\arctan(x)&=\int {\frac {dx}{1+x^{2}}}\quad &&{\text{in the range }}-\pi /2<\arctan(x)<\pi /2,\\&=\int {\frac {dx}{1-(-x^{2})}}\quad &&{\text{by writing integrand as closed form of geometric series with }}r=-x^{2},\\&=\int \left(1+\left(-x^{2}\right)+\left(-x^{2}\right)^{2}+\left(-x^{2}\right)^{3}+\cdots \right)dx\quad &&{\text{by writing geometric series in expanded form}},\\&=\int \left(1-x^{2}+x^{4}-x^{6}+\cdots \right)dx\quad &&{\text{by calculating the sign and power of each term in integrand}},\\&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots \quad &&{\text{by integrating each term}},\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}\quad &&{\text{by writing series in generator form}},\end{aligned}}$

which is called Gregory's series and is commonly attributed to Madhava of Sangamagrama (c. 1340 – c. 1425).

See also

• 0.999... – Alternative decimal expansion of 1
• Asymptote – Limit of the tangent line at a point that tends to infinity
• Divergent geometric series
• Generalized hypergeometric function – Family of power series in mathematics
• Geometric progression – Mathematical sequence of numbers
• Neumann series – Mathematical series
• Ratio test – Criterion for the convergence of a series
• Root test – Criterion for the convergence of an infinite series
• Series (mathematics) – Infinite sum
• Arithmetic series – Sequence of equally spaced numbersPages displaying short descriptions of redirect targets

References

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History and philosophy

• C. H. Edwards Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0-387-94313-8.
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• Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN 978-0-691-02511-7
• Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0-415-22526-7

Economics

• Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN 978-0-393-95733-4
• Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7

Biology

• Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0-387-09648-3
• Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press. ISBN 978-0-521-57698-7

Computer science

• John Rast Hubbard (2000). Schaum's Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0-07-137870-3

External links

• "Geometric progression", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Geometric Series". MathWorld.
• Geometric Series at PlanetMath.
• Peppard, Kim. "College Algebra Tutorial on Geometric Sequences and Series". West Texas A&M University.
• Casselman, Bill. "A Geometric Interpretation of the Geometric Series". Archived from the original (Applet) on 2007-09-29.
• "Geometric Series" by Michael Schreiber, Wolfram Demonstrations Project, 2007.