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This article is about infinite geometric series. For finite sums, see 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
The sum of the areas of the purple squares is one third of the area of the large square.
The sum of the areas of the purple squares is one third of the area of the large square.

In mathematics, a geometric series is a series with a constant ratio between successive terms. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" For example, the series

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

is geometric, because each term is equal to half of the previous term. The sum of this series is 1, as illustrated in the following picture:

Geometric series are the simplest examples of infinite series with finite sums. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with This makes them important in philosophy, where they provide a mathematical resolution to Zeno's paradoxes. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, and finance. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Foundations of modern biology There are five unifying principles Economics is the social science that studies the production distribution, and consumption of goods and services. The field of finance refers to the concepts of Time, Money and Risk and how they are interrelated

Contents

Common ratio

The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found 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:

When r is greater than one, the terms of the series become larger and larger.
When r is less than one (and greater than zero), the terms of the series become smaller and smaller, approaching zero in the limit.
When r is equal to one, all of the terms of the series are the same.

The common ratio can also be negative, which causes the sign of the terms to alternate. A negative number is a Number that is less than zero, such as −2

Sum

The sum of a geometric series is finite as long as the terms approach zero. The sum can be computed using the self-similarity of the series. In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i

Example

A self-similar illustration of the sum s. Removing the leftmost circle has the same effect as shrinking the picture by a factor of 1/3.
A self-similar illustration of the sum s. In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i Removing the leftmost circle has the same effect as shrinking the picture by a factor of 1/3.

Consider the sum of the following geometric series:

s \;=\; 1 \,+\, \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \cdots

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:

\frac{2}{3}s \;=\; \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \frac{16}{81} \,+\, \cdots

This new series is the same as the original, except that the first term is missing. Subtracting the two series cancels every term but the first:

s \,-\, \frac{2}{3}s \;=\; 1,\;\;\;\;\;\;\;\;\mbox{so }s=3.

A similar technique can be used to evaluate any self-similar expression. In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i

Formula

There is a general formula for the sum of a geometric series: \sum_{k=0}^{n-1} ar^k=a\frac{1-r^n}{1-r} (r not equal to 1). As n goes to infinity, the absolute value of 'r' must be less than one for the series to converge. The series then becomes

s \;=\; \frac{a}{1-r}.

Here a is the first term of the series, and r is the common ratio. When a = 1, this simplifies to the following formula

1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \;=\; \frac{1}{1-r},

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

\begin{array}{l} \text{Let }s \;=\; 1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots. \\[4pt] \text{Then }rs \;=\; r \,+\, r^2 \,+\, r^3 \,+\, r^4 \,+\, \cdots. \\[4pt] \text{Then }s \,-\, rs \;=\; 1,\;\;\;\;\mbox{so }s\,=\,\frac{1}{1-r}. \end{array}

The general formula follows if we multiply through by a.

This formula is only valid for convergent series (i. e. when the magnitude of r is less than one). In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. For example, the sum is undefined when r = 10, even though the formula gives s = –1/9.

This reasoning is also valid, with the same restrictions, for the complex case. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Proof of convergence

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

\begin{align} &1 \,+\, r \,+\, r^2 \,+\, r^3 \,+\, \cdots \\[3pt] &=\; \lim_{n\rightarrow\infty} \left(1 \,+\, r \,+\, r^2 \,+\, \cdots \,+\, r^n\right) \\ &=\; \lim_{n\rightarrow\infty} \frac{1-r^{n+1}}{1-r} \end{align}

Since rn+1 → 0 for | r | < 1, the limit is 1 / (1 – r). In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found

Representation using sigma notation

Using sigma notation, a geometric series with common ratio r and first term a can be written as follows:

\sum_{n=0}^\infty ar^n

It is important to begin the summation at n = 0, for this makes the first term ar0 = a.

Applications

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. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is For example:

0.7777\ldots \;=\; \frac{7}{10} \,+\, \frac{7}{100} \,+\, \frac{7}{1000} \,+\, \frac{7}{10,000} \,+\, \cdots.

You can use the formula for the sum of a geometric series to convert the decimal to a fraction:

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

Archimedes' quadrature of the parabola

The area enclosed by a parabola and a line is the union of infinitely many triangles.
The area enclosed by a parabola and a line is the union of 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. The Quadrature of the Parabola is a treatise on Geometry, written by Archimedes in the 3rd century B Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular His method was to dissect the area into an infinite number of triangles, as shown in the figure to the right.

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, 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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position

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. The sum is

\frac{1}{1-r}\;=\;\frac{1}{1-\frac{1}{4}}\;=\;\frac{4}{3}.    Q. E. D.

This computation uses the method of exhaustion, an early version of integration. 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 The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In modern calculus, the same area could be found using a definite integral. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

Fractal geometry

The interior of the Koch snowflake is the union of infinitely many triangles.
The interior of the Koch snowflake is the union of infinitely many triangles. The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described

In the study of fractals, geometric series often arise as the perimeter, area, or volume of a self-similar figure. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" The perimeter is the distance around a given two-dimensional object Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In Mathematics, a self-similar object is exactly or approximately similar to a part of itself (i

For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles (see figure). The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described Properties The area of an equilateral triangle with sides of length a\\! 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.

Zeno's paradoxes

Main article: Zeno's paradoxes

Understanding the convergence of a geometric series allows to resolve many of Zeno's paradoxes as it reveals that a sum of an infinite set can remain finite for | r | < 1. For example Zeno's dichotomy paradox attains that movement is impossible, as one can divide any path into steps of one half of the distance remaining, thus an infinite number of steps is needed to cross any finite distance. The hidden assumption is that a sum of infinite number of finite steps can not be finite. This is of course not true as evident by the convergence of the geometrical series with r=1/2 illustrated at the picture at the introduction section of this article.


Euclid

Book IX, Proposition 35 of Euclid's Elements expresses the partial sum of a geometric series in terms of members of the series. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek It is equivalent to the modern formula.

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). The time value of money is based on the premise that an Investor prefers to receive a payment of a fixed amount of money today rather than an equal amount in the future Economics is the social science that studies the production distribution, and consumption of goods and services. Present 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

For example, suppose that you expect to receive a payment of $100 once per year in perpetuity. A perpetuity is an annuity that has no definite end or a stream of cash payments that continues forever Receiving $100 a year from now is worth less to you than an immediate $100, because you cannot invest the money until you receive it. Investment or investing is a term with several closely-related meanings in Business management, Finance and Economics, related to saving 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:

\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

\frac{a}{1-r} \;=\; \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 $1000.

This sort of calculation is used to compute the APR of a loan (such as a mortgage). Annual percentage rate (APR is the simplified counterpart to the Effective interest rate that the borrower will pay on a loan A mortgage is the pledging of a property to a Lender as a security for a Mortgage loan. It can also be used to estimate the present value of expected stock dividends, or the terminal value of a security. Dividends are payments made by a Corporation to its Shareholder members A security is a Fungible, Negotiable instrument representing financial value

Geometric power series


See also

Specific geometric series

References

History and philosophy

Economics

Biology

Computer science

External links

QUESTION. Find the smallest number in a GP whose sum is 38 and product 1728

Dictionary

geometric series

-noun

  1. (analysis) Infinite series whose terms are in a geometric progression.
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