Lecture

Linear Recurrence Sequences

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Description

This lecture covers the concept of linear recurrence sequences, where a sequence is defined by a linear formula involving the previous terms. The instructor explains the theorem and the process of defining such sequences, emphasizing the role of initial conditions. The lecture delves into examples and properties of these sequences, discussing convergence and divergence criteria. The instructor demonstrates the convergence of sequences towards a limit and the conditions for divergence. Additionally, the lecture explores the concept of Cauchy sequences and their relevance in analyzing the behavior of linear recurrence sequences.

In MOOCs (2)

Analysis I (part 3) : Real number sequences I and II

A sequence of real numbers is a function f:N→R . It is usual to write an:=f(n) for the value of f in n. For example, we could define a sequence f(n):=an:=12n, i.e. a0=1,a1=12,a2=14,a3=18,.... . The central concept is that of the limit of a sequence: this is a real number to which, intuitively, the given sequence gets closer and closer. For example, the sequence an given above admits the number zero as its limit. We'll define the concept of limit in a rigorous way and develop methods for establishing the exi

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 (27)

Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position.

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become arbitrarily close to the term. For instance, in the sequence of square roots of natural numbers: the consecutive terms become arbitrarily close to each other – their differences tend to zero as the index n grows.

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Cauchy product

In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section.

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as if (and only if) The function is said to be the pointwise limit function of the Sometimes, authors use the term bounded pointwise convergence when there is a constant such that .

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