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Home lessons in mathematics and computer science and physics
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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