Analysis III: Integration and Elementary Differential Equations

Building on the analysis course from the second semester, this course will cover the concepts of series with terms of arbitrary sign. The Riemann integral will be defined and applied to solve differential equations, particularly linear ones. The section on integration will be expanded to include generalized integrals.

Series with terms of any sign

• Cauchy criterion, absolute convergence
• Other convergence criteria: Leibniz’s rule (for alternating series) and Abel’s rule

- Use of DLs to prove convergence.

- Analysis of the remains, convergence rate.

 Integration

- Integral of a step function

- Integrable Riemann Functions

- Primitives and Integrals

- Some calculation methods (IPP, substitution, formulas for the mean)

- Riemann sums

 Differential Equations

- Equations with separable variables

- First-Order Linear Equations

- Second-order linear equations (with constant coefficients).

- Nonlinear equations (Ricatti, Bernoulli)

Generalized Integrals

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

- The Cauchy criterion.

- Comparisons of generalized integrals with positive terms.

- Absolute convergence criteria.

- Semi-convergent integrals.

HAX201X – Analysis II: Sequences, Series, and Limited Expansions

Recommended prerequisites: First-year math

Hourly volumes:

            CM: 30

            TD: 30

            Practical Work:

            Lot: