Geometric series

Geometric series

Overview

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a geometric series is a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

with a constant ratio between successive terms
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...

. For example, the series

is geometric, because each successive 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 have this property.
Discussion

Encyclopedia

In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a geometric series is a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

with a constant ratio between successive terms
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...

. For example, the series

is geometric, because each successive 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 have this property. Historically, geometric series played an important role in the early development of calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...

, and finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

.

Common ratio

The terms of a geometric series form a geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

, meaning that the ratio of successive terms in the series is constant. The following table shows several geometric series with different common ratios:
Common ratio Example
10 4 + 40 + 400 + 4000 + 40,000 + ···
1/3 9 + 3 + 1 + 1/3 + 1/9 + ···
1/10 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ···
1 3 + 3 + 3 + 3 + 3 + ···
−1/2 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ···
–1 3 − 3 + 3 − 3 + 3 − ···

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 the limit and the series converges to a sum. In the case above, where r is one half, the series has the sum one.
If r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges
Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit....

.)
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 terms oscillates
Oscillation (mathematics)
In mathematics, oscillation is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; that is, oscillation is the failure to have a limit, and is also a quantitative measure for that.Oscillation is defined as the...

between two values (e.g. 2, 0, 2, 0, 2,... ). This is a different type of divergence and again the series has no sum. See for example Grandi's series: 1 − 1 + 1 − 1 + ···.

Sum

The sum
SUM
SUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...

of a geometric series is finite as long as the terms approach zero; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series being infinite. The sum can be computed using the self-similarity
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

of the series.

Example

Consider the sum of the following geometric series:
This series has common ratio 2/3. If we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on:
This new series is the same as the original, except that the first term is missing. Subtracting the new series (2/3)s from the original series cancels every term in the original but the first:
A similar technique can be used to evaluate any self-similar expression.

Formula

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

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

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

When , this simplifies to:

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

The general formula follows if we multiply through by a.

This formula is only valid for convergent series (i.e., when the magnitude
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of r is less than one). For example, the sum is undefined when , even though the formula gives .

This reasoning is also valid, with the same restrictions, for the complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

case.

Proof of convergence

We can prove that the geometric series converges using the sum formula for a geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

:
Since (1 + r + r2 + ... + rn)(1−r) = 1−rn+1 and for | r | < 1, the limit is .

Repeating decimals

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

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

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

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

Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...

used the sum of a geometric series to compute the area enclosed by a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

and a straight line. His method was to dissect the area into an infinite number of triangles.

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

Proof: Using his extensive knowledge of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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

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

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

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

The sum is    Q.E.D.

This computation uses the method of exhaustion
Method of exhaustion
The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will...

, an early version of integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

. In modern calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

, the same area could be found using a definite integral.

Fractal geometry

In the study of fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

s, geometric series often arise as the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...

, area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

, or volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

of a self-similar
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself . Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales...

figure.

For example, the area inside the Koch snowflake
Koch snowflake
The Koch snowflake is a mathematical curve and one of the earliest fractal curves to have been described...

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

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

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

The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...

's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein each step is taken to be half the remaining distance. Zeno's mistake is in the assumption that the sum of an infinite number of finite steps cannot be finite. This is of course not true, as evidenced by the convergence of the geometric series with .

Euclid

Book IX, Proposition 35 of Euclid's Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...

expresses the partial sum of a geometric series in terms of members of the series. It is equivalent to the modern formula.

Economics

In economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, geometric series are used to represent the present value
Present value
Present value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...

of an annuity
Annuity (finance theory)
The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money...

(a sum of money to be paid in regular intervals).

For example, suppose that you expect to receive a payment of \$100 once per year (at the end of the year) in perpetuity
Perpetuity
A perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence...

. Receiving \$100 a year from now is worth less to you than an immediate \$100, because you cannot invest
Investment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...

the money until you receive it. In particular, the present value of a \$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 it would have received the yearly interest twice). Therefore, the present value of receiving \$100 per year in perpetuity

can be expressed as an infinite series:

This is a geometric series with common ratio 1 / (1 + I). The sum is

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

This sort of calculation is used to compute the APR
Annual percentage rate
The term annual percentage rate , also called nominal APR, and the term effective APR, also called EAR, describe the interest rate for a whole year , rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate...

of a loan (such as a mortgage loan
Mortgage loan
A mortgage loan is a loan secured by real property through the use of a mortgage note which evidences the existence of the loan and the encumbrance of that realty through the granting of a mortgage which secures the loan...

). It can also be used to estimate the present value of expected stock dividends
Dividend
Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...

, or the terminal value
Terminal value
Terminal value can mean several things:*In accounting, terminal value refers to the salvage or residual value of an asset.*In computer science, terminal value refers to the character that signify the end of a line....

of a security
Security (finance)
A security is generally a fungible, negotiable financial instrument representing financial value. Securities are broadly categorized into:* debt securities ,* equity securities, e.g., common stocks; and,...

.

Geometric power series

The formula for a geometric series

can be interpreted as a power series in the Taylor's theorem
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial. For analytic functions the Taylor polynomials at a given point are finite order truncations of its Taylor's series, which completely determines the...

sense, converging where . From this, one can extrapolate to obtain other power series. For example,

• asymptote
Asymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

• series (mathematics)
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

• geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

• ratio test
• root test
• divergent geometric series
• Neumann series
• Tower of Hanoi
Tower of Hanoi
The Tower of Hanoi or Towers of Hanoi, also called the Tower of Brahma or Towers of Brahma, is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod...

• 0.999...
0.999...
In mathematics, the repeating decimal 0.999... denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number...

Specific geometric series

• Grandi's series: 1 − 1 + 1 − 1 + · · ·
• 1 + 2 + 4 + 8 + · · ·
1 + 2 + 4 + 8 + · · ·
In mathematics, 1 + 2 + 4 + 8 + … is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so in the usual sense it has no sum...

• 1 − 2 + 4 − 8 + · · ·
• 1/2 + 1/4 + 1/8 + 1/16 + · · ·
• 1/2 − 1/4 + 1/8 − 1/16 + · · ·
• 1/4 + 1/16 + 1/64 + 1/256 + · · ·
1/4 + 1/16 + 1/64 + 1/256 + · · ·
In mathematics, the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. Its sum is 1/3...

History and philosophy

• C. H. Edwards, Jr. (1994). The Historical Development of the Calculus, 3rd ed., Springer. ISBN 978-0387943138.
• Eli Maor
Eli Maor
Eli Maor, an Israel-born historian of mathematics, is the author of several books about the history of mathematics. Eli Maor received his PhD at the Technion – Israel Institute of Technology. He teaches the history of mathematics at Loyola University Chicago...

(1991). To Infinity and Beyond: A Cultural History of the Infinite, Princeton University Press. ISBN 978-0691025117
• Morr Lazerowitz (2000). The Structure of Metaphysics (International Library of Philosophy), Routledge. ISBN 978-0415225267

Economics

• Carl P. Simon and Lawrence Blume (1994). Mathematics for Economists, W. W. Norton & Company. ISBN 978-0393957334
• Mike Rosser (2003). Basic Mathematics for Economists, 2nd ed., Routledge. ISBN 978-0415267847

Biology

• Edward Batschelet (1992). Introduction to Mathematics for Life Scientists, 3rd ed., Springer. ISBN 978-0387096483
• Richard F. Burton (1998). Biology by Numbers: An Encouragement to Quantitative Thinking, Cambridge University Press. ISBN 978-0521576987

Computer science

• John Rast Hubbard (2000). Schaum's Outline of Theory and Problems of Data Structures With Java, McGraw-Hill. ISBN 978-0071378703