Analysis III integration and differential equations element

Following on from the analysis course in S2, this course deals with the notion of series with terms of any sign. The Riemann integral will be defined and applied to linear and other differential equations. The integration section will be extended to generalized integrals.

Series with terms of any sign

• Cauchy criterion, absolute convergence
• other convergence criteria: Leibniz (alternating series) and Abel's rules

- use of DLs to prove convergence.

- study of remainders, speed of convergence.

 Integration

- Integral of a stepped function

- Integrable Riemann functions

- Primitives and Integrals

- Some calculation methods (PPI, change of variables, average formulas)

- Riemann sums

 Differential equations

- Equations with separable variables

- Linear of order 1

- Linear of order 2 (with constant coefficients).

- Non-linear equations (Ricatti, Bernoulli)

Generalized integrals

- Definitions: generalized convergent, absolutely convergent, semi-convergent and divergent integrals.

- The Cauchy criterion.

- Comparisons of generalized integrals with positive terms.

- Absolute convergence criteria.

- Semi-convergent integrals.

HAX201X - Analysis II Sequences, series, limited developments

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 30

            TD : 30

            TP :

            Terrain :