Analysis II Sequences, series, limited developments

This follows on from S1 (Analysis I), where continuity and differentiability of real functions, common functions, and the study of real sequences were introduced.

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

- Digital suites:

• Comparison relation on sequences (small o, big O, equivalent)
• Upper limit/lower limit, concept of adhesion value, Cauchy sequence (example of a Cauchy sequence of rational numbers that does not converge in Q)
• Bolzano-Weierstrass theorem.
• Study of recurring sequences (_un+1=f(_un))

- Actual 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: boundedness theorem, uniform continuity, Lipschitz 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, and in particular Analysis I, Reasoning and Set Theory, and Calculus or Remediation.

Recommended prerequisites:

S1 mathematics program.

Hourly volumes:

            CM: 30 hours

            Tutorial: 30 hours

            TP: 0

            Land: 0