**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Geometric series

Summary

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series
is geometric, because each successive term can be obtained by multiplying the previous term by . In general, a geometric series is written as , where is the coefficient of each term and is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential.
The name geometric succession indicates each term is the geometric mean of its two neighboring terms, similar to how the name Arithmetic succession indicates each term is the arithmetic mean of its two neighboring terms.
The geometric series a + ar + ar2 + ar3 + ... is written in expanded form. Every coefficient in the geometric series is the same. In contrast, the power series written as a0 + a1r + a2r2 + a3r3 + ... in expanded form has coefficients ai that can vary from term to term. In other words, the geometric series is a special case of the power series. The first term of a geometric series in expanded form is the coefficient a of that geometric series.
In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as
and a closed form of the geometric series written as
The derivation of the closed form from the expanded form is shown in this article's section. However even without that derivation, the result can be confirmed with long division: a divided by (1 - r) results in a + ar + ar2 + ar3 + ... , which is the expanded form of the geometric series.
It is often a convenience in notation to set the series equal to the sum s and work with the geometric series
s = a + ar + ar2 + ar3 + ar4 + ... in its normalized form
s / a = 1 + r + r2 + r3 + r4 + ... or in its normalized vector form
s / a = [1 1 1 1 1 ...][1 r r2 r3 r4 .

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (44)

Related people (2)

Related concepts (27)

Related courses (27)

Related lectures (104)

Series (mathematics)

In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Taylor series

In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-101(e): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-207(d): Analysis IV

Le cours étudie les concepts fondamentaux de l'analyse complexe et de l'analyse de Laplace en vue de leur utilisation
pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

Convergence and Cauchy Sequences

Explores convergence and Cauchy sequences, including the Bolzano-Weierstrass theorem and the properties of convergent sequences.

Geometric Expansion of Crom-Conrelation

Explores the geometric expansion of the crom-conrelation, focusing on mathematical formulas and correlation analysis.

Convergence of Series

Explores the convergence criteria of numerical series, including absolute convergence and alternating series.

Victor Panaretos, Yoav Zemel, Valentina Masarotto

We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an Analysis of Vari ...

Self-propelled particles such as bacteria or algae swimming through a fluid are non-equilibrium systems where particle motility breaks microscopic detailed balance, often resulting in large-scale collective motion. Previous theoretical work has identified ...

Corentin Jean Dominique Fivet, Tao Sun

This paper presents a geometry-driven approach to form-finding with reused stock elements. Our proposed workflow uses a K-mean algorithm to cluster stock elements and incorporate their geometrical values early in the form-finding process. A feedback loop i ...