Continuum Mathematics

Complex numbers Algebraic and trigonometric representations, algebraic operations and roots of unity. The Fundamental Theorem of Algebra, factoring of polynomials.
Real numbers, rational numbers and integers. Comparison between rational and irrational numbers: countable and uncountable sets. Maximum and minimal elements of subsets of the real line, greatest lower bound and least upper bound.
Natural numbers: Induction over the integers and properties that holds eventually.
Sequences of real numbers: basic properties, boundedness and monotonicity.
Limits of sequences: the notion of limit, uniqueness, boundedness of convergent sequences, comparison theorems. Algebraic operations with limits and forms of indecision, comparison between infinite/infinitesimal sequences, ratio and root tests, Landau's symbol little-o and its use, the concept of asymptotic equivalence and its use. Regularity of monotone sequences, the number e (of Napier). Other Landau symbols: big-o, big-omega, big-theta and their use in the comparison of sequences.
Continuous functions: the notion of continuity and its graphical interpretation, points of discontinuity. The concept of limit for functions and its relation to continuity. Continuity and discontinuity of elementary functions: rational, exponential, logarithmic functions, the absolute value and step functions, the integer and fractional part functions. Change of variables in limits and the limit of compositions of functions. The theorem on zeros and Weirstrass' theorem for continuous functions.
Differential calculus: the concept of the derivative: linear approximations and the tangent to a curve. Calculation of derivatives for elementary functions. Angular points and cusps. Algebraic operations and the derivative. The theorems of Fermat, Rolle, Lagrange and applications. Cauchy's theorem. De l'Hôpital's theorem and comparison on the order of infinity. The formula of Taylor and its applications. Optimization problems (finding maxima and minima). Integral calculus: computing areas, approximation and the method of exhaustion. The integral of Riemann: definition of the definite integral, classes of integrable functions, properties of the definite integral.
The mean value theorem for integrals and the fundamental theorem of calculus.
Indefinite integrals and their calculation: integration by substitution and by parts,
integration of rational functions. Improper integrals: definition and fundamental examples.
Finite sums: shifts, inresions and other algebraic manipulations. Fundamental examples: powers of integers and geometric progressions.
The concept of series: fundamental examples, the geometric series and telescopic series.
Necessary condition for convergence, regularity, comparison test, absolute convergence and simple convergence. Estimates and asymptotic estimates on the rate of conver-gence/divergence of a series: comparison test, limit comparison test. The generalized harmonic series (p-series).
Real power series and the radius of convergence. Taylor series and analytic functions. Algebraic operations on power series, derivative and integrals of series.
Introduction to recursion: problems of counting and recurrence equations. Solving recurrence equations by the method of the (ordinary) generating functions: formal manipulations of power series.