Geometric Series

Geometric seriesFrom Wikipedia, the free encyclopedia

Contents

1 Geometric progression 11.1 Elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Related formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Infinite geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Relationship to geometry and Euclid’s work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Geometric series 92.1 Common ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.3 Proof of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Generalized formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Repeating decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Archimedes’ quadrature of the parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Fractal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Zeno’s paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.5 Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.6 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.7 Geometric power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 Specific geometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.1 History and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.2 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

i

ii CONTENTS

2.5.3 Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.4 Computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Chapter 1

Geometric progression

Diagram illustrating three basic geometric sequences of the pattern 1(rn−1) up to 6 iterations deep. The first block is a unit block andthe dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3respectively.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers whereeach term after the first is found by multiplying the previous one by a fixed, non-zero number called the commonratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5,2.5, 1.25, ... is a geometric sequence with common ratio 1/2.Examples of a geometric sequence are powers rk of a fixed number r, such as 2k and 3k. The general form of ageometric sequence is

a, ar, ar2, ar3, ar4, . . .

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence’s start value.

1

2 CHAPTER 1. GEOMETRIC PROGRESSION

1.1 Elementary properties

The n-th term of a geometric sequence with initial value a and common ratio r is given by

an = a rn−1.

Such a geometric sequence also follows the recursive relation

an = r an−1 for every integer n ≥ 1.

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in thesequence all have the same ratio.The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbersswitching from positive to negative and back. For instance

1, −3, 9, −27, 81, −243, ...

is a geometric sequence with common ratio −3.The behaviour of a geometric sequence depends on the value of the common ratio.If the common ratio is:

• Positive, the terms will all be the same sign as the initial term.

• Negative, the terms will alternate between positive and negative.

• Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign ofthe initial term).

• 1, the progression is a constant sequence.

• Between −1 and 1 but not zero, there will be exponential decay towards zero.

• −1, the progression is an alternating sequence

• Less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alter-nating sign.

Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, asopposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with commondifference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields ageometric progression, while taking the logarithm of each term in a geometric progression with a positive commonratio yields an arithmetic progression.An interesting result of the definition of a geometric progression is that for any value of the common ratio, any threeconsecutive terms a, b and c will satisfy the following equation:

b2 = ac

where b is considered to be the geometric mean between a and c.

1.2. GEOMETRIC SERIES 3

1.2 Geometric series

Computation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted fromthe original sequence. Two terms remain: the first term, a, and the term one beyond the last, or arm. The desiredresult, 312, is found by subtracting these two terms and dividing by 1 − 5.A geometric series is the sum of the numbers in a geometric progression. For example:

2 + 10 + 50 + 250 = 2 + 2× 5 + 2× 52 + 2× 53.

Letting a be the first term (here 2), m be the number of terms (here 4), and r be the constant that each term ismultiplied by to get the next term (here 5), the sum is given by:

a(1− rm)

1− r

In the example above, this gives:

2 + 10 + 50 + 250 =2(1− 54)

1− 5=

−1248

−4= 312.

The formula works for any real numbers a and r (except r = 1, which results in a division by zero). For example:

−2π + 4π2 − 8π3 = −2π + (−2π)2 + (−2π)3 =−2π(1− (−2π)3)

1− (−2π)=

−2π(1 + 8π3)

1 + 2π≈ −214.855.

1.2.1 Derivation

To derive this formula, first write a general geometric series as:

n∑k=1

ark−1 = ar0 + ar1 + ar2 + ar3 + · · ·+ arn−1.

We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll seethat

(1− r)n∑

k=1

ark−1 = (1− r)(ar0 + ar1 + ar2 + ar3 + · · ·+ arn−1)

= ar0 + ar1 + ar2 + ar3 + · · ·+ arn−1 − ar1 − ar2 − ar3 − · · · − arn−1 − arn

= a− arn

since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometricseries that computes the sum of n terms:

n∑k=1

ark−1 =a(1− rn)

1− r.

1.2.2 Related formulas

If one were to begin the sum not from k=0, but from a different value, say m, then

4 CHAPTER 1. GEOMETRIC PROGRESSION

n∑k=m

ark =a(rm − rn+1)

1− r.

Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form

n∑k=0

ksrk.

For example:

d

dr

n∑k=0

rk =n∑

k=1

krk−1 =1− rn+1

(1− r)2− (n+ 1)rn

1− r.

For a geometric series containing only even powers of r multiply by 1 − r2 :

(1− r2)

n∑k=0

ar2k = a− ar2n+2.

Then

n∑k=0

ar2k =a(1− r2n+2)

1− r2.

Equivalently, take r2 as the common ratio and use the standard formulation.For a series with only odd powers of r

(1− r2)n∑

k=0

ar2k+1 = ar − ar2n+3

and

n∑k=0

ar2k+1 =ar(1− r2n+2)

1− r2.

1.2.3 Infinite geometric series

Main article: Geometric series

An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series convergesif and only if the absolute value of the common ratio is less than one (|r| < 1). Its value can then be computed fromthe finite sum formulae

∞∑k=0

ark = limn→∞

n∑k=0

ark = limn→∞

a(1− rn+1)

1− r=

a

1− r− lim

n→∞

arn+1

1− r

Since:

1.2. GEOMETRIC SERIES 5

1

1/2

1/41/8

1/161/32

1/641/128

Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + which converges to 2.

rn+1 → 0 as n → ∞ when |r| < 1.

Then:

∞∑k=0

ark =a

1− r− 0 =

a

1− r

For a series containing only even powers of r ,

∞∑k=0

ar2k =a

1− r2

and for odd powers only,

∞∑k=0

ar2k+1 =ar

1− r2

In cases where the sum does not start at k = 0,

∞∑k=m

ark =arm

1− r

The formulae given above are valid only for |r| < 1. The latter formula is valid in every Banach algebra, as long as thenorm of r is less than one, and also in the field of p-adic numbers if |r|p < 1. As in the case for a finite sum, we candifferentiate to calculate formulae for related sums. For example,

d

dr

∞∑k=0

rk =∞∑k=0

krk−1 =1

(1− r)2

This formula only works for |r| < 1 as well. From this, it follows that, for |r| < 1,

6 CHAPTER 1. GEOMETRIC PROGRESSION

∞∑k=0

krk =r

(1− r)2 ;

∞∑k=0

k2rk =r (1 + r)

(1− r)3 ;

∞∑k=0

k3rk =r(1 + 4r + r2

)(1− r)

4

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ⋯ is an elementary example of a series that converges absolutely.It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is

1

2+

1

4+

1

8+

1

16+ · · · = 1/2

1− (+1/2)= 1.

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series thatconverges absolutely.It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is

1

2− 1

4+

1

8− 1

16+ · · · = 1/2

1− (−1/2)=

1

3.

1.2.4 Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. Inthis case the condition that the absolute value of r be less than 1 becomes that the modulus of r be less than 1. It ispossible to calculate the sums of some non-obvious geometric series. For example, consider the proposition

∞∑k=0

sin(kx)rk

=r sin(x)

1 + r2 − 2r cos(x)

The proof of this comes from the fact that

sin(kx) = eikx − e−ikx

2i,

which is a consequence of Euler’s formula. Substituting this into the original series gives

∞∑k=0

sin(kx)rk

=1

2i

[ ∞∑k=0

(eix

r

)k

−∞∑k=0

(e−ix

r

)k]

This is the difference of two geometric series, and so it is a straightforward application of the formula for infinitegeometric series that completes the proof.

1.3 Product

The product of a geometric progression is the product of all terms. If all terms are positive, then it can be quicklycomputed by taking the geometric mean of the progression’s first and last term, and raising that mean to the powergiven by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence:take the arithmetic mean of the first and last term and multiply with the number of terms.)

∏ni=0 ar

i =(√

a0 · an)n+1 (if a, r > 0 ).

1.4. RELATIONSHIP TO GEOMETRY AND EUCLID’S WORK 7

Proof:Let the product be represented by P:

P = a · ar · ar2 · · · arn−1 · arn

Now, carrying out the multiplications, we conclude that

P = an+1r1+2+3+···+(n−1)+(n)

Applying the sum of arithmetic series, the expression will yield

P = an+1rn(n+1)

2

P = (arn2 )n+1

We raise both sides to the second power:

P 2 = (a2rn)n+1 = (a · arn)n+1

Consequently

P 2 = (a0 · an)n+1

P = (a0 · an)n+12

which concludes the proof.

1.4 Relationship to geometry and Euclid’s work

Books VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the articlefor details) and give several of their properties.[1]

1.5 See also

• Arithmetic progression

• Arithmetico-geometric sequence

• Exponential function

• Harmonic progression

• Harmonic series

• Infinite series

• Preferred number

• Thomas Robert Malthus

• Geometric distribution

8 CHAPTER 1. GEOMETRIC PROGRESSION

1.6 References[1] • Heath, Thomas L. (1956). The Thirteen Books of Euclid’s Elements (2nd ed. [Facsimile. Original publication:

Cambridge University Press, 1925] ed.). New York: Dover Publications.

• Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2

1.7 External links• Hazewinkel, Michiel, ed. (2001), “Geometric progression”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Derivation of formulas for sum of finite and infinite geometric progression at Mathalino.com

• Geometric Progression Calculator

• Nice Proof of a Geometric Progression Sum at sputsoft.com

• Weisstein, Eric W., “Geometric Series”, MathWorld.

Chapter 2

Geometric series

This article is about infinite geometric series. For finite sums, see geometric progression.In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the

1/2

1/2

1/4

1/4

1/81/8

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 theareas of the purple squares is one third of the area of the large square.

series

9

10 CHAPTER 2. GEOMETRIC SERIES

1

2+

1

4+

1

8+

1

16+ · · ·

is geometric, because each successive term can be obtained by multiplying the previous term by 1/2.Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them havethis property. Historically, geometric series played an important role in the early development of calculus, and theycontinue to be central in the study of convergence of series. Geometric series are used throughout mathematics, andthey have important applications in physics, engineering, biology, economics, computer science, queueing theory,and finance.

2.1 Common ratio

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

1

1/2

1/4

1/8

1/161/32

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

The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the seriesis constant. This relationship allows for the representation of a geometric series using only two terms, r and a. Theterm r is the common ratio, and a is the first term of the series. As an example the geometric series given in theintroduction,

12 + 1

4 + 18 + 1

16 + · · ·

2.2. SUM 11

may simply be written as

a+ ar + ar2 + ar3 + · · · , with r = 12 and a = 1

2 .

The following table shows several geometric series with different common ratios:The behavior of the terms depends on the common ratio r:

If r is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in thelimit and the series converges to a sum. In the case above, where r is one half, the series has the sumone.If r is greater than one or less than minus one the terms of the series become larger and larger inmagnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The seriesdiverges.)If r is equal to one, all of the terms of the series are the same. The series diverges.If r is minus one the terms take two values alternately (e.g. 2, −2, 2, −2, 2,... ). The sum of the termsoscillates between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again theseries has no sum. See for example Grandi’s series: 1 − 1 + 1 − 1 + ···.

2.2 Sum

The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers nearzero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely-manyterms. The sum can be computed using the self-similarity of the series.

2.2.1 Example

A self-similar illustration of the sum s. Removing the largest circle results in a similar figure of 2/3 the original size.

Consider the sum of the following geometric series:

s = 1 +2

3+

4

9+

8

27+ · · ·

This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the2/3 becomes a 4/9, and so on:

2

3s =

2

3+

4

9+

8

27+

16

81+ · · ·

12 CHAPTER 2. GEOMETRIC SERIES

This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s fromthe original series s cancels every term in the original but the first:

s − 2

3s = 1, so s = 3.

A similar technique can be used to evaluate any self-similar expression.

2.2.2 Formula

For r ̸= 1 , the sum of the first n terms of a geometric series is:

a+ ar + ar2 + ar3 + · · ·+ arn−1 =n−1∑k=0

ark = a1− rn

1− r,

where a is the first term of the series, and r is the common ratio. We can derive this formula as follows:

Lets = a+ ar + ar2 + ar3 + · · ·+ arn−1.

Thenrs = ar + ar2 + ar3 + ar4 + · · ·+ arn

Thens− rs = a− arn

Thens(1− r) = a(1− rn), so s = a1− rn

1− r(ifr ̸= 1).

As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes

a+ ar + ar2 + ar3 + ar4 + · · · =∞∑k=0

ark =a

1− r, for |r| < 1.

When a = 1, this can be simplified to:

1 + r + r2 + r3 + · · · =1

1− r,

the left-hand side being a geometric series with common ratio r. We can derive this formula:

Lets = 1 + r + r2 + r3 + · · · .

Thenrs = r + r2 + r3 + · · · .

Thens− rs = 1, so s(1− r) = 1, thus and s = 1

1− r.

The general formula follows if we multiply through by a.The formula holds true for complex “r”, with the same restrictions (modulus of “r” is strictly less than one).

2.2.3 Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression:

1 + r + r2 + r3 + · · · = limn→∞

(1 + r + r2 + · · ·+ rn

)= lim

n→∞

1− rn+1

1− r

2.2. SUM 13

Since (1 + r + r2 + ... + rn)(1−r) = 1−rn+1 and rn+1 → 0 for | r | < 1.Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series.Consider the function:

g(K) =rK

1− r

Note that:

1 = g(0)− g(1), r = g(1)− g(2), r2 = g(2)− g(3), · · ·

Thus:

S = 1 + r + r2 + r3 + ... = (g(0)− g(1)) + (g(1)− g(2)) + (g(2)− g(3)) + · · ·

If

|r| < 1

then

g(K) −→ 0 asK → ∞

So S converges to

g(0) =1

1− r.

2.2.4 Generalized formula

For r ̸= 1 , the sum of the first n terms of a geometric series is:

b∑k=a

rk =ra − rb+1

1− r,

where a, b ∈ N .We can derive this formula as follows:we put b = n− 1 ⇒ n = b+ 1

b∑k=a

rk =n−1∑k=0

rk −a−1∑k=0

rk

=1− rn

1− r− 1− ra

1− r

=1− rn − 1 + ra

1− r

=ra − rb+1

1− r

14 CHAPTER 2. GEOMETRIC SERIES

2.3 Applications

2.3.1 Repeating decimals

Main article: Repeating decimal

A repeating decimal can be thought of as a geometric series whose common ratio is a power of 1/10. For example:

0.7777 . . . =7

10+

7

100+

7

1000+

7

10000+ · · · .

The formula for the sum of a geometric series can be used to convert the decimal to a fraction:

0.7777 . . . =a

1− r=

7/10

1− 1/10=

7

9.

The formula works not only for a single repeating figure, but also for a repeating group of figures. For example:

0.123412341234 . . . =a

1− r=

1234/10000

1− 1/10000=

1234

9999.

Note that every series of repeating consecutive decimals can be conveniently simplified with the following:

0.09090909 . . . =09

99=

1

11.

0.143814381438 . . . =1438

9999.

0.9999 . . . =9

9= 1.

That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part (as an integer) and 10n- 1.

2.3.2 Archimedes’ quadrature of the parabola

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. Hismethod 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 thearea of a green triangle, and so forth.Assuming that the blue triangle has area 1, the total area is an infinite sum:

1 + 2

(1

8

)+ 4

(1

8

)2

+ 8

(1

8

)3

+ · · · .

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

1 +1

4+

1

16+

1

64+ · · · .

2.3. APPLICATIONS 15

Archimedes’ dissection of a parabolic segment into infinitely many triangles

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

∞∑n=0

4−n = 1 + 4−1 + 4−2 + 4−3 + · · · = 4

3.

The sum is

1

1− r=

1

1− 14

=4

3.

This computation uses the method of exhaustion, an early version of integration. In modern calculus, the same areacould be found using a definite integral.

2.3.3 Fractal geometry

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure.For example, 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 thereforehas exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking theblue triangle as a unit of area, the total area of the snowflake is

1 + 3

(1

9

)+ 12

(1

9

)2

+ 48

(1

9

)3

+ · · · .

The first term of this series represents the area of the blue triangle, the second term the total area of the three greentriangles, the third term the total area of the twelve yellow triangles, and so forth. Excluding the initial 1, this seriesis 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

16 CHAPTER 2. GEOMETRIC SERIES

The interior of the Koch snowflake is a union of infinitely many triangles.

1 +a

1− r= 1 +

13

1− 49

=8

5.

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

2.3.4 Zeno’s paradoxes

Main article: Zeno’s paradoxes

The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed befinite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno’s dichotomy paradox maintains

2.3. APPLICATIONS 17

that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step istaken to be half the remaining distance. Zeno’s mistake is in the assumption that the sum of an infinite number offinite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series withr = 1/2 .

2.3.5 Euclid

Book IX, Proposition 35[1] of Euclid’s Elements expresses the partial sum of a geometric series in terms of membersof the series. It is equivalent to the modern formula.

2.3.6 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 inregular intervals).For example, suppose that a payment of $100 will be made to the owner of the annuity once per year (at the end ofthe year) in perpetuity. Receiving $100 a year from now is worth less than an immediate $100, because one cannotinvest 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 twoyears’ worth of interest is lost by not receiving the money right now). Therefore, the present value of receiving $100per year in perpetuity is

∞∑n=1

$100

(1 + I)n,

which is the infinite series:

$100

(1 + I)+

$100

(1 + I)2+

$100

(1 + I)3+

$100

(1 + I)4+ · · · .

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

$100/(1 + I)

1− 1/(1 + I)=

$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 estimatethe present value of expected stock dividends, or the terminal value of a security.

2.3.7 Geometric power series

The formula for a geometric series

1

1− x= 1 + x+ x2 + x3 + x4 + · · ·

can be interpreted as a power series in the Taylor’s theorem sense, converging where |x| < 1 . From this, one canextrapolate to obtain other power series. For example,

18 CHAPTER 2. GEOMETRIC SERIES

tan−1(x) =

∫dx

1 + x2

=

∫dx

1− (−x2)

=

∫ (1 +

(−x2

)+(−x2

)2+(−x2

)3+ · · ·

)dx

=

∫ (1− x2 + x4 − x6 + · · ·

)dx

= x− x3

3+

x5

5− x7

7+ · · ·

=∞∑

n=0

(−1)n

2n+ 1x2n+1

By differentiating the geometric series, one obtains the variant[2]

∞∑n=1

nxn−1 =1

(1− x)2for |x| < 1.

Similarly obtained are:

∞∑n=2

n(n− 1)xn−2 =2

(1− x)3for |x| < 1,

∞∑n=3

n(n− 1)(n− 2)xn−3 =6

(1− x)4for |x| < 1.

2.4 See also• 0.999...

• Asymptote

• Divergent geometric series

• Generalized hypergeometric function

• Geometric progression

• Neumann series

• Ratio test

• Root test

• Series (mathematics)

• Tower of Hanoi

2.4.1 Specific geometric series

• Grandi’s series: 1 − 1 + 1 − 1 + ⋯

• 1 + 2 + 4 + 8 + ⋯

• 1 − 2 + 4 − 8 + ⋯

2.5. REFERENCES 19

• 1/2 + 1/4 + 1/8 + 1/16 + ⋯

• 1/2 − 1/4 + 1/8 − 1/16 + ⋯

• 1/4 + 1/16 + 1/64 + 1/256 + ⋯

2.5 References

[1] “Euclid’s Elements, Book IX, Proposition 35”. Aleph0.clarku.edu. Retrieved 2013-08-01.

[2] Taylor, Angus E. (1955), Advanced Calculus, Blaisdell, p. 603

• Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

• Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 278-279, 1985.

• Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.

• Courant, R. and Robbins, H. “The Geometric Progression.” §1.2.3 in What Is Mathematics?: An ElementaryApproach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 13-14, 1996.

• Pappas, T. “Perimeter, Area & the Infinite Series.” The Joy of Mathematics. San Carlos, CA: Wide WorldPubl./Tetra, pp. 134-135, 1989.

• James Stewart (2002). Calculus, 5th ed., Brooks Cole. ISBN 978-0-534-39339-7

• Larson, Hostetler, and Edwards (2005). Calculus with Analytic Geometry, 8th ed., Houghton Mifflin Company.ISBN 978-0-618-50298-1

• Roger B. Nelsen (1997). Proofs without Words: Exercises in Visual Thinking, The Mathematical Associationof America. ISBN 978-0-88385-700-7

• Andrews, George E. (1998). “The geometric series in calculus”. The American Mathematical Monthly (Math-ematical Association of America) 105 (1): 36–40. doi:10.2307/2589524. JSTOR 2589524.

2.5.1 History and philosophy

• C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0-387-94313-8.

• Swain, Gordon and Thomas Dence (April 1998). “Archimedes’ Quadrature of the Parabola Revisited”. Math-ematics Magazine 71 (2): 123–30. doi:10.2307/2691014. JSTOR 2691014.

• Eli Maor (1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN978-0-691-02511-7

• Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN978-0-415-22526-7

2.5.2 Economics

• Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN978-0-393-95733-4

• Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0-415-26784-7

20 CHAPTER 2. GEOMETRIC SERIES

2.5.3 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 Uni-versity Press. ISBN 978-0-521-57698-7

2.5.4 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

2.6 External links• Hazewinkel, Michiel, ed. (2001), “Geometric progression”, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-4

• Weisstein, Eric W., “Geometric Series”, MathWorld.

• Geometric Series at PlanetMath.org.

• Peppard, Kim. “College Algebra Tutorial on Geometric Sequences and Series”. West Texas A&MUniversity.

• Casselman, Bill. “A Geometric Interpretation of the Geometric Series” (Applet).

• “Geometric Series” by Michael Schreiber, Wolfram Demonstrations Project, 2007.

2.7. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 21

2.7 Text and image sources, contributors, and licenses

2.7.1 Text• Geometric progression Source: https://en.wikipedia.org/wiki/Geometric_progression?oldid=670950573 Contributors: AxelBoldt, Zun-

dark, Tarquin, Patrick, Michael Hardy, Delirium, Conti, Charles Matthews, Dcoetzee, Hyacinth, Fredrik, Matt me, R3m0t, May-ooranathan, Henrygb, Aetheling, Tobias Bergemann, Giftlite, Knutux, LucasVB, MarkSweep, Rpchase, Jcw69, Allefant, Moxfyre, MikeRosoft, MuDavid, Paul August, Aranel, Touriste, Elementalish, Aisaac, Msh210, Rh~enwiki, Arthena, PAR, Mosesofmason, Justinle-bar, Olethros, Gerbrant, Graham87, Yurik, Sango123, Lmatt, BradBeattie, Chobot, Sbrools, Redde, Siddhant, YurikBot, Wavelength,Icedemon, JabberWok, Dantheox, DarthVader, Haihe, Plamka, EAderhold, Lt-wiki-bot, Arthur Rubin, Nemu, Mike1024, Pred, Hearth,Banus, Thorney¿?, Finell, SmackBot, RDBury, Incnis Mrsi, Melchoir, Nereus124, Ixtli, Janmarthedal, Bluebot, Octahedron80, DHN-bot~enwiki, Can't sleep, clown will eat me, Jratt, Mark Wolfe, Nakon, Stefano85, Vina-iwbot~enwiki, Netnubie, Jim.belk, Advance512,Mets501, Pjrm, JForget, CmdrObot, Ichiroo, FilipeS, Haifadude, ST47, Goldencako, Tawkerbot4, Joeyfox10, Awmorp, Vanished Userjdksfajlasd, Thijs!bot, Dugwiki, AntiVandalBot, Михајло Анђелковић, JAnDbot, Leuko, Divyesikka, Gaeddal, 01001, MSBOT, James-BWatson, Meissmart, JJ Harrison, David Eppstein, Quanticle, Andylatto, Chrisalvino, Policron, DavidCBryant, DorganBot, Gp4rts,VolkovBot, Johan1298~enwiki, Jeff G., LokiClock, Philip Trueman, Af648, TXiKiBoT, Vertciel, SieBot, Yulu, Anchor Link Bot,Timeastor, ClueBot, Justin W Smith, DR23, Mathwizkid, He7d3r, Fattyjwoods, NellieBly, Addbot, Laubpatr, Metagraph, Fieldday-sunday, Tide rolls, Jarble, Odder, Luckas-bot, Yobot, გიგა, AnomieBOT, The Parting Glass, 9258fahsflkh917fas, Xqbot, Resident Mario,Krishano, LuisVillegas, Joxemai, RTFVerterra, Pepper, Gas Panic42, Amgc56, Turian, FoxBot, TobeBot, Thelema418, Nascar1996,Skittlestastegood, Jowa fan, EmausBot, Felix Hoffmann, Wikipelli, Tahdah, Slawekb, Josve05a, L Kensington, Maschen, ClueBot NG,Wcherowi, PoqVaUSA, Ghostsarememories, Sparkie82, Rahaven, Brad7777, Anbu121, Arpitkjain, Henri.vanliempt, Amirki, 1Minow,Brirush, Dennis at Empa Media, Jhncls, Zereth, *thing goes, Hazo11413 and Anonymous: 221

• Geometric series Source: https://en.wikipedia.org/wiki/Geometric_series?oldid=672472757 Contributors: AxelBoldt, Bryan Derksen,The Anome, XJaM, Heron, Michael Hardy, Willsmith, Pnm, ArnoLagrange, LittleDan, Poor Yorick, Jitse Niesen, Hyacinth, Henrygb,Per Abrahamsen, Vacuum, Giftlite, MSGJ, Vsb, Moxfyre, Rich Farmbrough, Guanabot, Paul August, Touriste, Kenyon, Mindma-trix, Salix alba, The wub, Nihiltres, Kri, Wavelength, Gillis, Closedmouth, Arthur Rubin, Reyk, Netrapt, Ghazer~enwiki, SmackBot,Michaelliv, Incnis Mrsi, InverseHypercube, Golwengaud, Kostmo, Hgrosser, Jbergquist, Black Carrot, Jim.belk, Happy-melon, Lavat-eraguy, CBM, Schaber, Arrataz, Gogo Dodo, Escarbot, Uplink3r, Thenub314, JamesBWatson, JJ Harrison, David Eppstein, JaGa,Ankitdoshi1, Eastmbr, R'n'B, Pbroks13, AstroHurricane001, Policron, Fylwind, Austinmohr, Pleasantville, LokiClock, Philip Trueman,Ocolon, Chenzw, StevenJohnston, Oboeboy, Caltas, Yerpo, T5j6p9, Archaeogenetics, Khvalamde, Shane87, Asperal, PerryTachett, Dr-garden, DonAByrd, ClueBot, Justin W Smith, DanielDeibler, Timberframe, Niceguyedc, Hans Adler, BOTarate, Eranus~enwiki, PCHS-NJROTC, HiTechHiTouch, Addbot, DOI bot, Zarcadia, Jarble, Clay Juicer, Yobot, AnomieBOT, Bdmy, Dithridge, Trut-h-urts man,Raffamaiden, NOrbeck, Hugetim, Efadae, MrHeberRomo, Citation bot 1, S iliad, Hexadecachoron, Duoduoduo, Thelema418, Bobby122,WillNess, Ramblagir, Slawekb, 4blossoms, Souless194, VoilàY'all, DASHBotAV, Mastomer, Rocketrod1960, ClueBot NG, Wcherowi,Helpful Pixie Bot, Jakemymath, Rahaven, Brad7777, ולדמן שמחה ,יהודה Zetazeros, OceanEngineerRI, Amirki, Webclient101, Saehry,Stephan Kulla, Frosty, Doctordubin, Hillbillyholiday, CsDix, Gkvp, Babitaarora, ColeLoki, Bellezzasolo, Staymathy, Vrkssai, Monkbot,Ktlabe, Tymon.r, Feitreim and Anonymous: 166

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