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280 private teachers in Nigeria

Léon - Douala£15
£15Home lessons in mathematics and computer science and physics
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
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Koffi - Akropong£21
£21Learn French in a simple, easy and fun way. Let's embark on this journey now.
Chat in French, Read & Write French, Sing in French the simple, easy, and fun way. Bonjour! Bonjour!; Comment allez-vous? Quand a moi. je vais super bien. Hello! Hello! How are you doing? As for me, I am super fine. If you are planning to travel or settle in any Francophone countries for business purposes or for work matters...Or you just want to learn French to polish and sharpen your academic or take your professional career to the next level, you are at the right place at the right time. I am called Koffi affectionately nicknamed Tamos As you can see, it might sound difficult to pronounce my name in the first place but with the pronunciation, you can easily and confidently say it without any worry. Yes, pronunciation is key if you want to speak any language as a native speaker. French is not an exception. This is one of the most important aspects of learning French. It will help me understand and to be understood. Four months ago, I started learning German as my fourth language. Very hard. My teachers are trying their best to teach me the correct pronunciation and I pick up slowly but surely. This is my background. I am born in Ghana - studied in Ghana, Togo, and Cote d’Ivoire. Currently, I am teaching French and English in an international, multicultural, multilingual school complex, and in a private nursing university on weekends, all in Ghana. I also do translations in my spare time. By the way, I am also a writer. Prior to that, I taught at Alliance Francaise Accra (a member of ALLIANCE FRANCAISE NETWORK) and got the opportunity to manage ACCENT LANGUAGE INSTITUTE where we teach both English and French to Francophones and Anglophones. I taught beginners, intermediate and advanced students. Some of my students are kids, youngsters, adults, professionals. . As a perfect speaker, my role as your French teacher will be to give you the perfect mentoring and coaching to make you improve and polish your French skills just like I did for many people in the past. I strongly believe that anyone has the cognitive abilities to learn a new language and become fluent as a native speaker. Maybe next year, I will be offering German lessons. My lessons are conversational, functional, and self-explanatory. They are easy, simple, and fun. It consists of a step-by-step approach or methods of teaching. These are things that you are going to learn through the four skills (speaking, listening, reading, and writing) are as follow: 1- CONVERSATIONS 2- DIALOGUES 3- SONGS 4- VOCABULARY 5- GRAMMAR 6- PRONUNCIATION 7- PROJECT-BASED LEARNING(PBL) MODE OF TEACHING VIDEO AUDIO TEXT IMAGES/PICTURES CANAL OF TEACHING ZOOM GOOGLE MEET WHATSAPP YOUTUBE FACEBOOK LIVE BONUSES FOR JOINING MY CLASS You will have access to my WHATSAAP GROUP You will have access to my FRENCH GROUP FOR ENGLISH SPEAKERS where you will exchange and network with peers and other learners. Just know why you want to learn French and I will help you through the process. Again feel free to contact me something above is not clear to you. A bientot Koffi
French
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