Lecture

Taylor Series of a Function

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Description

This lecture covers the Taylor series of a function, emphasizing the use of the Taylor formula for functions of class C² on an open interval. It explains the representation of a function through its Taylor series, also known as the Maclaurin series, and discusses the concept of remainder in the Taylor formula. The lecture delves into the application of the Taylor series with examples and clarifies the terminology related to the Taylor series. It concludes by highlighting the importance of the Taylor series in approximating functions and its significance in mathematical analysis.

In MOOCs (2)

Analysis I (part 6) : Study of functions, limited developments

The study of functions is the discussion of some of their properties. To do this, we need certain theorems that allow us, for example, to find the variations of the function under study. We've already seen the theorem of finite increments. In this chapter, we study its generalization. When we study a function, we also want to know how it behaves at infinity. Unfortunately, the limits of some functions can be quite complicated to study, so we introduce the Bernoulli-l'Hospital rule and demonstrate it. This r

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

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Related concepts (30)

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.

Colin Maclaurin

Colin Maclaurin (məˈklɔːrən; Cailean MacLabhruinn; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Maclaurin was born in Kilmodan, Argyll.

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation I is most commonly reserved for the closed interval .

Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).

Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.

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