Analysis II: Sequences, Series, and Finite Series

This course follows on from the S1 course (Analysis I), in which we were introduced to the continuity and differentiability of real-valued functions, common functions, and the study of real sequences.

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

- Numerical sequences:

• Comparison of Sequences (small o, big O, equivalent)
• Upper and lower bounds, the concept of adhesion, Cauchy sequences (example of a Cauchy sequence of rational numbers that does not converge in Q)
• Bolzano–Weierstrass Theorem.
• Study of Recurrent Sequences (_un+1=f(_un))

- Real functions:

• Comparison Relations (small o, big O, equivalent)
• Limited expansions and the Taylor-Lagrange formula, Taylor-Young formula, common limited expansions, operations, applications of limited expansions to limit calculations, common inequalities, relative position of a curve with respect to its tangent, asymptotic analysis
• Regularity of functions: the Boundedness Theorem, uniform continuity, Lipschitz functions, Heine's Theorem.

- Study of numerical series:

• Geometric and telescoping series: a simple case with explicit calculation of partial sums
• Positive series (comparison relation, Riemann series, Cauchy–d’Alembert criterion, condensation criterion, Bertrand series)

First-semester mathematics curriculum, specifically Analysis I, Logical Reasoning and Set Theory, and Calculus or Remedial Math.

Recommended prerequisites:

First Semester Mathematics Curriculum.

Hourly volumes:

            CM: 30 hours

            TD: 30 hours

            TP: 0

            Land: 0