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Peter Wittwer

Une suite de nombres réels est une fonction f:N→R . Il est habituel d'écrire an:=f(n) pour la valeur de f en n. Par exemple, on pourrait définir une suite f(n):=an:=12n, c'est-à-dire a0=1,a1=12,a2=14,a3=18,... . Le concept central est celui de la limite d'une suite : c'est un nombre réel auquel, intuitivement, la suite donnée s'approche de plus en plus. Par exemple la suite an donnée en haut admet comme limite le nombre zéro. Nous définirons le concept de la limite d'une manière rigoureuse et développerons des méthodes pour établir l'existence d'une limite. En plus, nous découvrirons un lien entre le concept de la limite et celui de l'infimum et du supremum d'un ensemble. Une application très importante des suites de nombres réels est le fait que chaque nombre réel peut être considéré comme la limite d'une suite de nombres rationnels. Nous verrons comment obtenir le nombre irrationnel racione de 5 comme limite d'une suite de nombres rationnels. Nos étudions le concept des suites de Cauchy et des suites définies par récurrence linéaire. Nous montrons certaines propriétés des suites définies par récurrence linéaire, en faisant en lien avec les suites de Cauchy. Nous nous intéressons aux limites des suites et des sous-suites, ce qui nous amène au théorème de Bolzano-Weierstrass. A l'aide des suites, nous définissons aussi le concept des séries numériques que nous illustrons à l'aide de différents exemples. Nous définissons certains critères de convergence pour les séries, notamment le critère de d'Alembert, le critère de Cauchy, le critère de comparaison et le critère de Leibniz. Finalement, nous étudions les séries numériques avec un paramètre.

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What's inside

Learning objectives

  • • concept des suites de nombres réels
  • • suites définies par récurrence
  • • limite d'une suite
  • • suites divergentes
  • • opérations algébriques sur les limites
  • • théorème des deux gendarmes
  • Critères de convergence
  • Convergence d'une suite définie par récurrence
  • Suites de cauchy
  • Construction de r (un modèle pour r)
  • Théorème de bolzano‐weierstrass
  • Limite inférieure et limite supérieure
  • Séries numériques

Syllabus

Chapitre 3 : Suites de nombres réels, I
3.1 Définitions et exemples
3.2 Suites définies par récurrence
3.3 Propriétés de base
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3.4 Limite d'une suite
3.5 Deux propositions
3.6 Suites divergentes
3.7 Opérations algébriques sur les limites
3.8 Théorème des deux gendarmes
3.9 Suites monotones
3.10 Convergence d'une suite définie par récurrence
3.11 Bon à savoir
Chapitre 4 : Suites de nombres réels, II
1.1 Les nombres rationnels, propriétés
1.2 Introduction axiomatique de R
1.3 Infimum
1.4 Supremum
1.5 Nombre réels, sqrt(2)
1.6 Sous‐ensembles de R
1.7 Valeur absolue
1.8 Propriétés additionnelles de R

Good to know

Know what's good
, what to watch for
, and possible dealbreakers
Développe des concepts avancés en mathématiques, ce qui est essentiel pour les professionnels des domaines techniques
Fournit une base solide pour comprendre les limites et les séries, ce qui est crucial pour les étudiants en mathématiques et les professionnels des domaines connexes
Enseigne des concepts fondamentaux en mathématiques, ce qui est bénéfique pour les étudiants souhaitant approfondir leurs connaissances dans ce domaine
Peut nécessiter un niveau intermédiaire en mathématiques, ce qui peut être un défi pour les débutants
Couvre des sujets avancés, ce qui peut être intimidant pour les étudiants peu familiers avec les mathématiques avancées

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Analyse I (partie 3) : Suites de nombres réels I et II with these activities:
Practice solving various types of equations
Staying sharp on solving equations is crucial for this course, and consistent practice can enhance your proficiency.
Browse courses on Equations
Show steps
  • Find practice problems online or in textbooks
  • Solve equations regularly
  • Check your answers
Organize your notes and materials on sequence convergence
By organizing your materials, you will be better prepared for this course's coverage of sequence convergence. This will help you achieve better learning outcomes in this course.
Browse courses on Limits
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  • Gather all of your notes, handouts, and other materials on sequence convergence.
  • Create a system for organizing your materials, such as a binder or folder.
  • Review your materials regularly to ensure that you understand the concepts.
Learn basic number theory and set theory
Reviewing these concepts will help you prepare for the course. Utilizing a guided tutorial can provide explicit and detailed instruction on these topics.
Browse courses on Set Theory
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  • Find tutorials on set theory and number theory
  • Take notes on important concepts
  • Complete practice problems
Seven other activities
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Show all ten activities
Review sequence convergence techniques
Complete exercises to prepare for this course's coverage of sequence convergence. This will help you achieve better learning outcomes in this course.
Browse courses on Limits
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  • Review the definition of a sequence and a limit.
  • Practice finding the limit of a sequence using the limit laws.
  • Try to prove the limit of a sequence using the epsilon-delta definition.
Review Differential Equations
This course is a continuation of Calculus, and specifically Calculus 2. Revisiting Differential Equations before the course begins will strengthen foundational knowledge and clear up any obstacles you may face later on.
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  • Review Chain Rule
  • Identify the types of Differential equations
  • Solve first and second order Differential equations
Discuss limits of sequences with peers
Engaging with other students will help you achieve better learning outcomes in this course. This activity will allow you to consolidate your understanding and prepare for this course's coverage of sequence convergence.
Browse courses on Limits
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  • Find a study partner or group.
  • Discuss the definition of a sequence and a limit.
  • Work together to solve practice problems on finding the limit of a sequence.
Practice problems on finding limits of sequences
Complete these drills to improve understanding of limits of sequences. By completing this, you will be preparing for this course's coverage of sequence convergence, and thus will achieve better learning outcomes in this course.
Browse courses on Limits
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  • Solve 10 practice problems on finding the limit of a sequence using the limit laws.
  • Solve 5 practice problems on proving the limit of a sequence using the epsilon-delta definition.
Exercices sur les limites
Pratiquez la résolution d'exercices sur les limites pour renforcer votre compréhension et développer votre dextérité.
Show steps
  • Résoudre des exercices sur les limites de fonctions polynomiales.
  • Appliquer les règles de l'algèbre des limites.
  • Calculer les limites de fonctions trigonométriques.
  • Résoudre des problèmes impliquant des limites infinies.
Write a summary of the different methods for finding the limit of a sequence
This activity will help you reinforce your understanding of the different methods for finding the limit of a sequence. This will help you achieve better learning outcomes in this course.
Browse courses on Limits
Show steps
  • Review your notes and textbooks on the different methods for finding the limit of a sequence.
  • Write a summary of the different methods, including examples.
  • Share your summary with your classmates or instructor for feedback.
Résumé des concepts clés
Créez un résumé écrit ou une présentation visuelle résumant les concepts clés du cours, tels que les limites, les suites et les théorèmes.
Show steps
  • Identifier les concepts clés abordés dans chaque chapitre.
  • Élaborer des définitions claires et des explications concises.
  • Inclure des exemples et des contre-exemples pour illustrer les concepts.
  • Parcourir le résumé pour garantir la compréhension et l'exactitude.

Career center

Learners who complete Analyse I (partie 3) : Suites de nombres réels I et II will develop knowledge and skills that may be useful to these careers:
Actuary
Actuaries analyze the financial consequences of risk and uncertainty. They use mathematical and statistical techniques to assess the probability of future events, and to develop strategies to manage risk. This course can provide a strong foundation for a career as an Actuary, as it covers the basics of probability and statistics, as well as the mathematical techniques used to analyze risk. Also taught are theoretical concepts of limits, which are important for understanding the behavior of financial markets.
Financial Analyst
Financial Analysts evaluate and make recommendations on investments. They use financial data and models to assess the performance of companies and industries, and to make recommendations on how to invest money. Those who wish to enter a career in financial analysis will benefit greatly from this course, which covers key concepts in probability, statistics, and calculus, which are essential for understanding financial markets.
Quantitative Analyst
Quantitative Analysts develop and use mathematical and statistical models to analyze financial data. They use these models to make predictions about future market movements, and to develop trading strategies. A strong foundation in mathematics and statistics is essential to success in this role, and this course will provide you with the necessary skills.
Data Analyst
Data Analysts collect, clean, and analyze data. They use statistical techniques to identify trends and patterns in data, and to make recommendations on how to improve business outcomes. This course provides a strong foundation in statistics and probability, which are essential for success in this role.
Market Researcher
Market Researchers conduct surveys and focus groups to collect data on consumer behavior. They use this data to understand the needs and wants of consumers, and to develop marketing strategies. This course provides a strong foundation in statistics and probability, which are essential for success in this role.
Software Engineer
Software Engineers design, develop, and test software applications. They use mathematical and statistical techniques to ensure that software is efficient and effective. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Operations Research Analyst
Operations Research Analysts use mathematical and statistical techniques to improve the efficiency and effectiveness of business operations. They use these techniques to analyze data, identify problems, and develop solutions. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Insurance Underwriter
Insurance Underwriters assess the risk of insuring individuals and businesses. They use mathematical and statistical techniques to analyze data, and to determine the likelihood of an event occurring. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Statistician
Statisticians collect, analyze, and interpret data. They use statistical techniques to identify trends and patterns in data, and to make predictions about the future. This course provides a strong foundation in statistics and probability, which are essential for success in this role.
Risk Manager
Risk Managers identify and assess risks, and develop strategies to manage those risks. They use mathematical and statistical techniques to analyze data, and to make recommendations on how to reduce risk. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Financial Planner
Financial Planners help individuals and families plan for their financial future. They use mathematical and statistical techniques to analyze financial data, and to make recommendations on how to save and invest money. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Economist
Economists study the production, distribution, and consumption of goods and services. They use mathematical and statistical techniques to analyze economic data, and to make predictions about the future. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Business Analyst
Business Analysts use data and analysis to identify and solve business problems. They use mathematical and statistical techniques to analyze data, and to make recommendations on how to improve business outcomes. This course provides a strong foundation in mathematics and statistics, which are essential for success in this role.
Teacher
Teachers develop and deliver lesson plans, and teach students in a variety of subjects. They use a variety of teaching methods, including lectures, discussions, and hands-on activities. This course could provide a supplemental source of knowledge on the subject area of Mathematics, which could be helpful to developing lesson plans.
Technical Writer
Technical Writers create and edit technical documentation, such as user manuals, technical reports, and white papers. They use a variety of writing styles, including technical, scientific, and business writing. This course can help teachers develop the writing skills and knowledge of a subject area, which are essential for success in this role.

Reading list

We've selected 15 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Analyse I (partie 3) : Suites de nombres réels I et II.
Ce livre fournit une introduction complète et rigoureuse à l'analyse réelle, y compris les sujets abordés dans le cours.
Ce livre est une référence avancée qui peut être utilisée pour un traitement plus approfondi des sujets abordés dans le cours.
Ce livre est un texte complet qui couvre une large gamme de sujets en analyse réelle, y compris les sujets abordés dans le cours.
Ce livre couvre les concepts fondamentaux de l'analyse réelle, y compris les limites, les dérivées et les intégrales.
Ce livre offre une approche accessible et intuitive du sujet, ce qui en fait un bon choix pour les débutants.
Ce livre est une introduction à la théorie de l'intégration, qui est un sujet avancé lié à l'analyse réelle.
Ce livre fournit une introduction à la théorie de la mesure, qui est un sujet avancé lié à l'analyse réelle.

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