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Lecture# Limits of Sequences

Description

This lecture covers the concept of limits of sequences, including the definition of convergence towards infinity, properties of infinite limits, and the rule of the two gendarmes. It also discusses the case of sequences that tend towards positive or negative infinity and the implications for the convergence of the sequences.

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Related concepts (60)

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

Infinity

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.

Sequence homology

Sequence homology is the biological homology between DNA, RNA, or protein sequences, defined in terms of shared ancestry in the evolutionary history of life. Two segments of DNA can have shared ancestry because of three phenomena: either a speciation event (orthologs), or a duplication event (paralogs), or else a horizontal (or lateral) gene transfer event (xenologs). Homology among DNA, RNA, or proteins is typically inferred from their nucleotide or amino acid sequence similarity.

Conserved sequence

In evolutionary biology, conserved sequences are identical or similar sequences in nucleic acids (DNA and RNA) or proteins across species (orthologous sequences), or within a genome (paralogous sequences), or between donor and receptor taxa (xenologous sequences). Conservation indicates that a sequence has been maintained by natural selection. A highly conserved sequence is one that has remained relatively unchanged far back up the phylogenetic tree, and hence far back in geological time.

Sequence alignment

In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are typically represented as rows within a matrix. Gaps are inserted between the residues so that identical or similar characters are aligned in successive columns.

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