Refresher in mathematics

A sequence definition

A sequence is an infinite ordered list of numbers indexed by natural numbers

Arithmetic sequence

We say that (Un) forms an arithmetic sequence if there exists a real number d called the common difference such that for all n appartenant à n : Un+1 = Un + d

Geometric sequence

We say that the sequence (Un) forms a geometric progression if r#0 exists the common ratio such that Un = Un x R

Mathematical Induction

Let P(n) be a property that de-

pends on n∈N. If:

1. Base case: P(0) is true (or P(1) depending on the context)

2. Inductive step: For all k∈N, if P(k) is true, then P(k+ 1) is true

Then P(n) is true for all n∈N.

Conclusion

Induction writing

For n … Let us prove by induction that : P(n)…

Base case : For n we have …

Inductive step : Let k be an integer and let us assume that P(n) is true, and let’s verify if it is true for P (n+1)

Conclusion : P(n) is true for the first term and verified for P (n+1)

Proofs

Case disjunction, Contraposition, Contradiction

Proof by CONTRAPOSITION

When trying to prove statments of the form if A then B we prove if not B then A

Proof by contradiction

Contradiction is argument of the form not p gives c, therefore p

How ?

1. Suppose not p is true

2. Get a contradiction like 0=1

3. Therefore p is true

Case disjunction

We consider case by case

Divergence to infinity

A sequence (un) diverges to +∞ if for any M ∈ R, there exists n0 ∈ N such that for all n ≥ n0: un ≥ M We write: limn→∞ un = +∞ or un → +∞.

Convergence

Let (un) be a sequence and let ℓ ∈ R. We say that (un) converges to ℓ if for any ε > 0, there exists n0 ∈ N such that for all n ≥ n0: |un − ℓ| ≤ε

Continuity

We write limx→a f(x) = ℓ if: For any ε > 0, there exists δ > 0 such that for any x ∈ A with |x − a| ≤ δ: |f(x) − ℓ| ≤ ε

Let ε > 0. We are looking for a value of δ > 0 such that if |x| ≤ δ, then |2x+1−1| ≤ ε. Let δ = ε 2 . Then if |x| ≤ δ, we have: |2x + 1 − 1| = |2x| = 2|x| ≤ 2δ = 2 · ε 2 = ε Therefore, f is continuous at 0.