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Geometric series

Index Geometric series

In mathematics, a geometric series is a series with a constant ratio between successive terms. [1]

56 relations: Annual percentage rate, Annuity, Archimedes, Area, Asymptote, Biology, Calculus, Computer science, Convergent series, Divergent geometric series, Divergent series, Dividend, Economics, Eli Maor, Engineering, Equilateral triangle, Euclid's Elements, Finance, Fractal, Generalized hypergeometric function, Geometric progression, Grandi's series, Integral, Investment, Koch snowflake, Mathematics, Method of exhaustion, Mortgage loan, Neumann series, Oscillation (mathematics), Parabola, Perimeter, Perpetuity, Physics, Power series, Present value, Queueing theory, Ratio test, Root test, Security (finance), Self-similarity, Series (mathematics), Summation, Taylor's theorem, Telescoping series, Term (logic), Terminal value (finance), Volume, Wolfram Demonstrations Project, Zeno of Elea, ..., 0.999..., 1 + 2 + 4 + 8 + ⋯, 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 + ⋯. Expand index (6 more) »

Annual percentage rate

The term annual percentage rate of charge (APR), corresponding sometimes to a nominal APR and sometimes to an effective APR (or EAPR), is the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc.

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An annuity is a series of payments made at equal intervals.

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Archimedes of Syracuse (Ἀρχιμήδης) was a Greek mathematician, physicist, engineer, inventor, and astronomer.

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Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

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In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.

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Biology is the natural science that studies life and living organisms, including their physical structure, chemical composition, function, development and evolution.

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Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus), is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

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Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

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Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

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Divergent geometric series

In mathematics, an infinite geometric series of the form is divergent if and only if | r | ≥ 1.

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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 finite limit.

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A dividend is a payment made by a corporation to its shareholders, usually as a distribution of profits.

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Economics is the social science that studies the production, distribution, and consumption of goods and services.

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Eli Maor

Eli Maor, an Israel-born historian of mathematics, is the author of several books about the history of mathematics.

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Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.

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Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal.

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Euclid's Elements

The Elements (Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

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Finance is a field that is concerned with the allocation (investment) of assets and liabilities (known as elements of the balance statement) over space and time, often under conditions of risk or uncertainty.

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In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

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Generalized hypergeometric function

In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation.

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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.

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Grandi's series

In mathematics, the infinite series 1 - 1 + 1 - 1 + \dotsb, also written \sum_^ (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703.

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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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In general, to invest is to allocate money (or sometimes another resource, such as time) in the expectation of some benefit in the future – for example, investment in durable goods, in real estate by the service industry, in factories for manufacturing, in product development, and in research and development.

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Koch snowflake

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical curve and one of the earliest fractal curves to have been described.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Method of exhaustion

The method of exhaustion (methodus exhaustionibus, or méthode des anciens) 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.

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Mortgage loan

A mortgage loan, or simply mortgage, is used either by purchasers of real property to raise funds to buy real estate, or alternatively by existing property owners to raise funds for any purpose, while putting a lien on the property being mortgaged.

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Neumann series

A Neumann series is a mathematical series of the form where T is an operator.

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Oscillation (mathematics)

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a sequence or function varies between its extreme values as it approaches infinity or a point.

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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

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A perimeter is a path that surrounds a two-dimensional shape.

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A perpetuity is an annuity that has no end, or a stream of cash payments that continues forever.

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Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

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Present value

In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation.

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Queueing theory

Queueing theory is the mathematical study of waiting lines, or queues.

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Ratio test

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series where each term is a real or complex number and is nonzero when is large.

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Root test

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.

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Security (finance)

A security is a tradable financial asset.

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In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

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Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

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In mathematics, summation (capital Greek sigma symbol: ∑) is the addition of a sequence of numbers; the result is their sum or total.

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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.

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Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a fixed number of terms after cancellation.

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Term (logic)

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact.

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Terminal value (finance)

In finance, the terminal value (continuing value or horizon value) of a security is the present value at a future point in time of all future cash flows when we expect stable growth rate forever.

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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

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Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

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Zeno of Elea

Zeno of Elea (Ζήνων ὁ Ἐλεάτης) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides.

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In mathematics, 0.999... (also written 0., among other ways), denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it).

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1 + 2 + 4 + 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two.

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1 − 2 + 4 − 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs.

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1/2 + 1/4 + 1/8 + 1/16 + ⋯

In mathematics, the infinite series is an elementary example of a geometric series that converges absolutely.

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1/2 − 1/4 + 1/8 − 1/16 + ⋯

In mathematics, the infinite series is a simple example of an alternating series that converges absolutely.

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1/4 + 1/16 + 1/64 + 1/256 + ⋯

In mathematics, the infinite series 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.

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Geometric Series, Geometric sum, Infinite geometric series.


[1] https://en.wikipedia.org/wiki/Geometric_series

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