Home lessons in mathematics and computer science and physics

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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

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Extra information

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Come with class lessons so that I can evolve the teacher of your school
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At student's location: Around Douala, Cameroon

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General info

Computer science

Math

Physics

Age:	Children (7-12 years old) Teenagers (13-17 years old) Adults (18-64 years old) Seniors (65+ years old)
Student level:	Beginner Intermediate Advanced
Duration:	120 minutes
The class is taught in:	French, English

About Me

I am a very understanding teacher and I fight for the student to understand what I teach him so that he is better in his establishment in the matter that I repeat, very rigorous on studies and I am relaxed when it is necessary to relax the atmosphere because too much stress prevents the student from thinking.

Education

Njombe High School, BEPC, 2014
Njombe High School, PROBATORY, 2016
Njombe High School, BACCALAUREATE C, 2015
University Institute of Technology, UNIVERSITY DIPLOMA OF TECHNOLOGY, 2017
University Institute of Technology, LICENSE OF TECHNOLOGY, 2018

Experience / Qualifications

2 years already in the repetitions of home courses in physical mathematics and computer science
All my students are always proud of my teachings

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Arithmetic for dummies in the final year

Arithmetic was initially limited to the study of the properties of natural integers, relative integers and rational numbers (in the form of fractions), and to the properties of operations on these numbers. The traditional arithmetic operations are addition, division, multiplication, and subtraction. This discipline was then broadened by the inclusion of the study of other numbers like reals (in the form of unlimited decimal expansion), or even more advanced concepts, like exponentiation or square root. Arithmetic is a way of formally representing - in other words, "coding" - numbers (as a list of digits, for example); and (thanks to this representation) define the basic operations: addition, multiplication, etc. Many integers have special properties. These properties are the subject of number theory. Among these particular numbers, the prime numbers are arguably the most important. Prime numbers edit This is the case for so-called prime numbers. These are the natural numbers having only two distinct positive divisors, namely 1 and themselves. The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. The integer 1 is not prime because it does not have two distinct positive divisors, but only one, namely himself. There are infinitely many prime numbers. By completing a grid of size 10 × 10 with the first 100 non-zero natural integers, and crossing out those which are not prime, we obtain the prime numbers belonging to {1, ..., 100} by a process called a sieve of Eratosthenes, named after the Greek scholar who invented it. Even and odd numbers edit Natural numbers can be divided into two categories: even and odd. An even {\ displaystyle n} n integer is a multiple of 2 and can therefore be written {\ displaystyle n = 2 \, k} n = 2 \, k, with {\ displaystyle k \ in \ mathbb {N} } k \ in \ N. An odd number {\ displaystyle n} n is not a multiple of 2 and can be written {\ displaystyle n = 2 \, k + 1} n = 2 \, k + 1, with {\ displaystyle k \ in \ mathbb {N}} k \ in \ N. We show that any integer is either even or odd, and this for a unique {\ displaystyle k} k: we denote {\ displaystyle \ forall n \ in \ mathbb {N} \ quad \ exists! K \ in \ mathbb {N } \ quad \ left (n = 2 \, k \ lor n = 2 \, k + 1 \ right)} {\ displaystyle \ forall n \ in \ mathbb {N} \ quad \ exists! k \ in \ mathbb { N} \ quad \ left (n = 2 \, k \ lor n = 2 \, k + 1 \ right)}. The first six even integers are 0, 2, 4, 6, 8 and 10. The first six odd integers are 1, 3, 5, 7, 9 and 11

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11	Monday at 11:00	Tuesday at 11:00	Wednesday at 11:00	Thursday at 11:00	Friday at 11:00	Saturday at 11:00	Sunday at 11:00
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