**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.

Lecture# Taylor Series: Convergence and Applications

Description

This lecture covers the concept of Taylor series, including its convergence properties and practical applications. It delves into the calculation of limits, composition, and product of Taylor series, as well as the representation of functions using Taylor series. The instructor explains the importance of understanding Taylor series in approximating functions and solving mathematical problems.

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.

In course

MATH-101(g): Analysis I

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

Instructor

Related concepts (139)

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. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series.

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.

Madhava series

In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are: All three series were later independently discovered in 17th century Europe.

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.

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

Related lectures (1,000)

Power Series and Taylor SeriesMATH-101(g): Analysis I

Explores power series, Taylor series, convergence criteria, and applications in mathematics.

Analysis I Exam SolutionsMATH-101(f): Analysis I

Provides solutions to an Analysis I exam, covering various topics.

Convergence of SeriesMATH-101(g): Analysis I

Covers the convergence criteria of series, including alternating series and absolute convergence.

Integration Techniques: Series and FunctionsMATH-101(g): Analysis I

Covers integration techniques for series and functions, including power series and piecewise functions.

Derivatives and FunctionsMATH-101(e): Analysis I

Covers the theorem of the intermediate value, corollaries, and geometric interpretation of derivatives.