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Trusted teacher: Digital suites courses I - General A numeric sequence is an application from N to R. • Bounded sequence A sequence (Un) is bounded if there exists a real A such that, for all n, Un ≤ A. We say that A is an upper bound of the series. A sequence (Un) is reduced if there exists a real number B such that, for all n, B ≤ one. One says that B is a lower bound of the sequence. A sequence is said to be bounded if it is both increased and reduced, that is to say if it exists M such that | Un | ≤ M for all n. • Convergent suite The sequence (Un) is convergent towards l ∈ R if: ∀ε> 0 ∃n0 ∈ N ∀n ≥ n0 | un − l | ≤ ε. A sequence which is not convergent is said to be divergent. When it exists, the limit of a sequence is unique. The deletion of a finite number of terms does not modify the nature of the sequence, nor its possible limit. Any convergent sequence is bounded. An unbounded sequence cannot therefore be convergent. • Infinite limits We say that the following (un) diverges Towards + ∞ if: ∀A> 0 ∃n0∈N ∀n ≥ n0 Un≥A Towards −∞ if: ∀A> 0 ∃n0∈N ∀n≤ n0 Un≤A. • Known limitations For k> 1, α> 0, β> 0 II Operations on suites • Algebraic operations If (un) and (vn) converge towards l and l ', then the sequences (un + vn), (λun) and (unvn) respectively converge towards l + l', ll and ll '. If (un) tends to 0 and if (vn) is bounded, then the sequence (unvn) tends to 0. • Order relation If (un) and (vn) are convergent sequences such that we have a ≤ vn for n≥n0, then we have: Attention, no analogous theorem for strict inequalities. • Framing theorem If, from a certain rank, un ≤xn≤ vn and if (un) and (vn) converge towards the same limit l, then the sequence (xn) is convergent towards l. III monotonous suites • Definitions The sequence (un) is increasing if un + 1≥un for all n; decreasing if un + 1≤un for all n; stationary if un + 1 = one for all n. • Convergence Any sequence of increasing and increasing reals converges. Any decreasing and underestimating sequence of reals converges. If a sequence is increasing and not bounded, it diverges towards + ∞. • Adjacent suites The sequences (un) and (vn) are adjacent if: (a) is increasing; (vn) is decreasing; If two sequences are adjacent, they converge and have the same limit. If (un) increasing, (vn) decreasing and un≤vn for all n, then they converge to l1 and l2. It remains to show that l1 = l2 so that they are adjacent. IV Extracted suites • Definition and properties - The sequence (vn) is said to be extracted from the sequence (un) if there exists a map φ of N in N, strictly increasing, such that vn = uφ (n). We also say that (vn) is a subsequence of (un). - If (un) converges to l, any subsequence also converges to l. If sequences extracted from (un) all converge to the same limit l, we can conclude that (un) converges to l if all un is a term of one of the extracted sequences studied. For example, if (u2n) and (u2n + 1) converge to l, then (un) converges to l. • Bolzano-Weierstrass theorem From any bounded sequence of reals, we can extract a convergent subsequence. V Suites de Cauchy • Definition A sequence (un) is Cauchy if, for any positive ε, there exists a natural integer n0 for which, whatever the integers p and q greater than or equal to n0, we have | up − uq | <ε. Be careful, p and q are not related. • Property A sequence of real numbers, or of complexes, converges if, and only if, it is Cauchy SPECIAL SUITES I Arithmetic and geometric sequences • Arithmetic sequences A sequence (un) is arithmetic of reason r if: ∀ n∈N un + 1 = un + r General term: un = u0 + nr. Sum of the first n terms: • Geometric sequences A sequence (un) is geometric of reason q ≠ 0 if: ∀ n∈N un + 1 = qun. General term: un = u0qn Sum of the first n terms: II Recurring suites • Linear recurrent sequences of order 2: - Such a sequence is determined by a relation of the type: (1) ∀ n∈N aUn + 2 + bUn + 1 + cUn = 0 with a ≠ 0 and c ≠ 0 and knowledge of the first two terms u0 and u1. The set of real sequences which satisfy the relation (1) is a vector space of dimension 2. We seek a basis by solving the characteristic equation: ar2 + br + c = 0 (E) - Complex cases a, b, c If ∆ ≠ 0, (E) has two distinct roots r1 and r2. Any sequence satisfying (1) is then like : where K1 and K2 are constants which we then express as a function of u0 and u1. If ∆ = 0, (E) has a double root r0 = (- b) / 2a. Any sequence satisfying (1) is then type: - Case a, b, c real If ∆> 0 or ∆ = 0, the form of the solutions is not modified. If ∆ <0, (E) has two conjugate complex roots r1 = α + iβ and r2 = α − iβ that we write in trigonometric form r1 = ρeiθ and r2 = ρe-iθ Any sequence satisfying (1) is then of the type: • Recurrent sequences un + 1 = f (un) - To study such a sequence, we first determine an interval I containing all the following values. - Possible limit If (un) converges to l and if f is continuous to l, then f (l) = l. - Increasing case f If f is increasing over I, then the sequence (un) is monotonic. The comparison of u0 and u1 makes it possible to know if it is increasing or decreasing. - Decreasing case f If f is decreasing over I, then the sequences (u2n) and (u2n + 1) are monotonic and of contrary Made by LEON
Math · Physics · Computer science
Computer science
Votre réussite en mathématiques, informatique, ou intelligence artificielle Besoin d'un coup de pouce dans une de ces matières ? Vous êtes au bon endroit ! Je vous propose des cours particuliers adaptés à votre niveau et vos objectifs, que vous soyez en collège, au lycée, ou que vous prépariez des études supérieures. Mes spécialités : - Fondamentaux des mathématiques : Algèbre, analyse, probabilités, statistiques, géométrie, et arithmétique. - Intelligence artificielle : Data science, machine learning, deep learning, optimisation, Large Language Models - Informatique : algorithmes, base de données, architecture, réseaux, avec python (numpy, pandas, pytorch, django), javascript, C, HTML/CSS, ou React - Physique : mécanique, électronique, thermodynamique, quantique Je comprends vos difficultés : Au fil de mes années d'expérience, j'ai identifié les principales difficultés rencontrées par les élèves en mathématiques : - Manque de bases solides : Je travaille à combler les lacunes pour vous donner une compréhension claire des concepts fondamentaux ainsi que des exemples simples pour développer votre intuition. - Difficulté à appliquer les théories : Je vous propose des exercices pratiques pour vous aider à maîtriser les méthodes de résolution de problèmes. - Peur des mathématiques : Je crée un environnement bienveillant où vous pouvez poser toutes vos questions sans jugement. C'est comme ça qu'on apprend. Ma méthode : -Évaluation personnalisée : Nous commençons par un bilan de vos connaissances et de vos objectifs pour définir un programme sur-mesure. -Enseignement interactif : Je vous propose des cours dynamiques et variés, avec des exercices corrigés en direct, des projets concrets et des mises en situation. - Suivi personnalisé : Vous bénéficiez d'un accompagnement régulier et de supports de cours exclusifs pour vous aider à progresser à votre rythme. Pourquoi me choisir ? - Double expertise : Mon expérience en recherche et dans l'industrie me permet de vous offrir une vision globale des mathématiques et de leurs applications. - Pédagogie innovante : Je vous propose une approche basée sur la résolution de problèmes réels pour rendre les mathématiques plus concrètes et plus intéressantes. - Réseau professionnel : Je peux vous mettre en relation avec des professionnels du secteur pour vous ouvrir de nouvelles perspectives. Les plus de mes cours : - Flexibilité : Les cours se déroulent en ligne ou en présentiel, à votre rythme et à votre convenance. - Personnalisation : Je m'adapte à votre profil et à vos objectifs pour vous garantir une progression optimale. - Résultats : De nombreux élèves ont pu améliorer leurs notes, surmonter leurs blocages et développer une véritable passion pour les mathématiques. Vous souhaitez en savoir plus ? N'hésitez pas à me contacter pour discuter de vos besoins et commencer votre parcours en mathématiques.
Math · Computer science · Physics
🌟✨ Discover Your Academic Edge with Jocelyne! ✨🌟 💻📚 Master IT, Computer Science, and IGCSE Subjects with an Expert! With a strong background in IT and Computer Science, I offer engaging and comprehensive lessons for students ranging from kids to high school. Whether you're exploring the basics of programming, tackling complex algorithms, or needing help with school subjects like Math, French, or Spanish, my sessions are designed to make learning effective and enjoyable. 👩‍💻💡 IT/Computer Science Expertise at Your Fingertips! As an IT professional holding a Bachelor's degree in Information Technology (Software Engineering) and currently pursuing a Master's in Computer Science at Aalborg University, I bring real-world experience and deep knowledge of coding languages like Python, C++, HTML, CSS, and JavaScript. My tutoring sessions include hands-on exercises, coding challenges, and problem-solving strategies tailored to your needs. 📘🎯 Personalized, Interactive Tutoring Approach! With my experience teaching Computer Science and IT to various age groups, I know how to tailor lessons that adapt to each student's learning pace. You can expect dynamic discussions, practical projects, and guidance that build both skill and confidence. 🔧🎓 Excel in Exams and Homework! Need support preparing for IGCSE exams or assistance with high school assignments? I’m here to help you succeed in Maths, English, Spanish, and other key subjects. My approach emphasizes clarity, interactive problem-solving, and strategic thinking. 🌱🌍 Join a Community of Learners! I believe in creating an environment that sparks curiosity and encourages growth. My tutoring sessions are structured to engage students, foster critical thinking, and inspire a love for learning. 🧠💪 Empower Your Academic Journey! Together, we will break down complex IT concepts, strengthen your coding skills, and set you on a path to academic success. 📚✨ Ready to advance your skills in IT and beyond? Let’s collaborate to turn your educational goals into accomplishments!
Information technology · Computer science · Tutoring
Trusted teacher: Hey! Are you looking for a young, approachable yet meticulous tutor in Computer Science or Mathematics? Then I believe I might be the tutor for you! I offer lessons in the aforementioned subjects (English only) for high-school and bachelor students at the start of their studies. A little bit about me: My name is Bozhidar and I am a MSc Data Science student at ETH Zuerich. Last year, I completed my bachelor's in Computer Science at TU Delft (the #1 technical university in the Netherlands) with distinction. I was a teaching assistant for 10 courses during my stay at the university. Additionally, I have given private lessons to a few high school and bachelor's students, so I believe I have accumulated enough experience with guiding and tutoring students across different disciplines. I have found out that I enjoy teaching a lot, so that is why I am here - to help you and myself! My expertise: During my high school years, I won tens of awards and scholarships for achievements in Mathematics and Computer Science. I also won my high school's recognition award which is given to one student every year for distinctive results. I've helped high-school and university peers with exam preparation, taught whole courses privately, prepared sample exams, and reviewed and graded real ones. Regarding the university courses I have experience in, I have taught: - Object-Oriented Programming - Reasoning & Logic - Computer Organisation - Algorithms and Data Structures - Web and Database Technologies - Calculus - Information & Data Management - Automata, Computability and Complexity - Concepts of Programming Languages - Computational Intelligence Here are some of the reviews I have received as a teaching assistant: Overall: 4.97/5 (70 reviews) 5/5 - "Great job! Did far more than just explaining this example and went far deeper into the material, helping me to understand it." 5/5 - "Favourite TA. Always there for the difficult questions. Never leaves anything unanswered no matter how long it takes." 5/5 - "Delightful person with an insightful way of thought! Brilliant TA!" 5/5 - "Clear explanations and expert in the material" So, what are you waiting for? Write me a message and let's get the knowledge train running.
Math · Computer science
Python · Computer science · Computer programming
Trusted teacher: Vous êtes impliqué dans les finances, la gestion, la planification, la gestion de projet, les ressources humaines, ou peut-être même un étudiant ambitieux désireux de perfectionner ses compétences en matière de tableau de bord professionnel. Si vous recherchez une solution qui va au-delà des limitations d'Excel et de PowerPoint, alors vous êtes au bon endroit : permettez moi de vous présenter Power BI. Avec Power BI, je vous propose bien plus qu'un simple outil. C'est une passerelle vers des rapports interactifs, une gestion efficace des données et une analyse avancée. Voici ce que je peux vous offrir : - Création et gestion experte de rapports interactifs. - Transformation et nettoyage minutieux des données pour une précision maximale. - Utilisation des puissantes formules DAX pour une analyse de données avancée. - Création de visualisations personnalisées et de tableaux de bord percutants. - Partage sécurisé et publication de vos rapports pour une collaboration sans heurts. -Automatisation des tâches répétitives avec Power BI & Power Query. Peu importe vos besoins spécifiques - qu'il s'agisse de projets professionnels, d'études ou d'aspirations personnelles - je suis là pour vous offrir une solution sur mesure. Ensemble, nous créerons un programme adapté à vos objectifs, vous guidant à chaque étape de votre parcours d'apprentissage. Que vous soyez un novice cherchant à maîtriser les bases ou un expert désireux d'approfondir vos connaissances en analyse de données, je suis là pour vous fournir l'expertise et le soutien nécessaires pour réussir.
Computer science · Microsoft excel
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Only reviews of students are published and they are guaranteed by Apprentus. Rated 4.8 out of 5 based on 88 reviews.

Mastering Web Development: From Fundamentals to Advanced Techniques (Tunis)
Oussama
I have no prior experience in this area and I took lessons with Ousamma. He has enough patience and a knack in guiding you step by step through the lessons and exercises in a very simple way to understand them well. He is an invested teacher that is kind and polite. He will help you to develop your abilities and to feel confident in order to achieve your goals. I warmly recommend him and extend my gratitude his great support throughout my learning journey.
Review by BEN
Support courses in mathematics intended for high school students, adapted to different sectors: Science Maths, Science Exp, Letters (Lyon)
Azzedine
Azzedine truly stands out as a remarkable instructor! Despite the short notice, he displayed remarkable consideration for our hectic timetable, ensuring to carve out time to equip me with Excel and Power BI skills for the assessment. A heartfelt thank you, Azzedine, for your exceptional teaching methods. I highly endorse Azzedine as your go-to teacher!
Review by IMANE
Tutor for Math, Physics, and Mechanical/Material Engineering courses (The Hague)
Reza
Extremely professional and knowledgeable with any problem that I have had. Reza is always polite, friendly and always shows great patience, which I believe is of the highest importance when learning difficult subjects. I highly recommend him as a teacher!
Review by EDEN