#### Objective

1. Knowledge of the notion of the simple and uniform convergence of a sequence of functions.
2. Knowledge of the criteria for uniform convergence.
3. Knowledge of the properties of the simple and uniform limit of a sequence of functions.
4. Knowledge of the properties of the uniform convergence on a closed bounded interval.
5. Know how to find the domain of convergence of a power series, the sum and the product of two power series.
6. Knowledge of the properties of the sum function of a power series in the interior of the domain of convergence : uniform convergence, uniform continuity, derivation and integration term by term.
7. Knowledge of the conditions and properties to express a function as the sum of a power series and the usual expansion.
8. Knowledge of the definition of a Fourier series of a numerical or complex function.
9. Knowledge of Euler formulas which permit the calculus of the Fourier series coefficients (case of odd and even functions).
10. Know how to give the complex form of the Fourier series.
11. Knowledge of the theorems about the convergence of the Fourier series of a numerical or complex function : Dirichlet theorem and Jordan theorem.
12. Find the sum of a Fourier series of a continuous or regular numerical function at a given point.
13. Have an idea about the trigonometric polynomials of approximation and in particular the Fourier polynomial.
14. Knowing how to handle the definite and generalized integrals depending upon a parameter ( continuity, derivation. summation, limit,…).

#### Content

• The space F(X,K) of functions of a set X with value in K(K = R ou C)
• Modes of convergence in F(X,K): Simple and uniform, compact for a metric space X..
• The space C(X) of functions from a metric space X into K.
• Dini theorem and Stone-Wieirstrass theorem. Applications.
• Series of functions defined in an open set of R or C : normal convergence,absolute convergence.
• Derivation et integration term by term.
• Real and complex Fourier series.
• Fourier series of a periodic function. Applications.