Analysis II Suites, series, limited developments

This course follows on from S1 (Analysis I), which introduced continuity and derivability of real functions, usual functions and the study of real sequences.

The aim is to continue and deepen work on sequences and functions, and to introduce the study of numerical series.

- Numerical sequences :

• Comparison relationship for sequences (small o, large O, equivalent)
• Limit sup/limit inf, notion of adhesion value, Cauchy sequence (example of a Cauchy sequence of rationals that does not converge in Q)
• Bolzano-Weierstrass theorem.
• Study of recurring sequences (_un+1=f(_un))

- Real functions :

• Comparison relationship (small o, large O, equivalent)
• Limited developments and Taylor-Lagrange formula, Taylor Young, usual limited developments, operations, applications of limited developments to limit calculations, usual inequalities, relative position of a curve with respect to its tangent, asymptotic study
• Regularity of functions: boundary theorem, uniform continuity, lipschitzian functions, Heine's theorem.

- Study of numerical series :

• Geometric and telescopic series, simple case with explicit calculation of partial sums
• Positive series (comparison relation, Riemann series, Cauchy/D'Alembert criterion, condensation criterion, Bertrand series)

S1 mathematics program, in particular Analysis I, Reasoning and set theory, and Calculus or Remediation.

Recommended prerequisites :

S1 mathematics program.

Hourly volumes :

            CM: 30 h

            TD: 30 h

            TP: 0

            Land: 0