**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 GraphSearch.

Concept# Taylor series

Summary

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.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a func

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

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (2)

Related publications (28)

Loading

Loading

Loading

Related units (1)

Francesca Bonizzoni, Fabio Nobile

We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theoretical upper bound for the norm of the residual of the Taylor expansion which predicts the divergence of the series as the polynomial degree goes to infinity. We provide a formula to compute the optimal degree for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach. Our theoretical findings are confirmed by numerical experiments in the simple case where the permeability field is described using one random variable

2014In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of $\infty$-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of $\infty$-operads and define tame spectral Lie algebras and tame spectral Hopf algebras. Finally, we prove that the homotopy theory of tame spectral Hopf algebras is equivalent to that of tame spaces. To recover Dwyer's Lie algebra model for tame spaces, we use Koszul duality to construct a universal enveloping algebra functor, and show it is an equivalence from the $\infty$-category of tame spectral Lie algebras to the $\infty$-category of tame spectral Hopf algebras.

Related courses (160)

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves.

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.

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.

Related concepts (112)

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate

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

Derivative

In mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the p

Related lectures (436)

Francesca Bonizzoni, Fabio Nobile

We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The series is directly computable in the case of a random field parametrized by a finite number of random variables. On the other hand, in the case of an infinite dimensional random field, suitable equations satisfied by the derivatives of the stochastic solution can be derived. We give a theoretical upper bound for the norm of the residual of the Taylor expansion which predicts the divergence of the series as the polynomial degree goes to infinity. We provide a formula to compute the optimal degree for the Taylor polynomial and the maximum achievable accuracy of the perturbation approach. Our theoretical findings are confirmed by numerical experiments in the simple case where the permeability field is described using one random variable.