Lectures related to Limits of Sequences: Induction, Bernoulli's Inequality, and Algebra

Limits of Sequences

Covers the concept of limits of sequences, including definitions, examples, boundedness, and more.

Limits to Infinity

Explores limits to infinity, including convergence, algebraic properties, and geometric progressions.

Real Analysis: Sequences and Limits

Covers real sequences, induction, limits, and convergence in mathematical analysis.

Subsequences and Bolzano-Weierstrass Theorem

Covers the proof of the Squeeze Theorem, Quotient Criteria, and the Bolzano-Weierstrass Theorem.

Modular Arithmetic: Foundations and Applications

Introduces modular arithmetic, its properties, and applications in cryptography and coding theory.

Lie Groups: Representations and Transformations

Explores Lie groups, scalar fields, and vector spaces transformations.

Polynomials: Definitions and Operations

Covers the definition and operations of polynomials, including addition and multiplication, degree, coefficients, and their role in algebraic systems.

Integration: more examples and rational functions

Covers antiderivatives, fundamental theorems, integration techniques, and rational functions.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Convergence of Fourier Series

Explores the convergence of Fourier series in L² space with trigonometric polynomials and approximation theorems.

Structure of Algebras

Covers the structure of finite dimensional algebras and the characterization of semisimple algebras.

Complement: Monotone Class Theorem

Explains the independence of events and sets in an algebraic context.

Banach Spaces: Reflexivity and Convergence

Explores Banach spaces, emphasizing reflexivity and sequence convergence in a rigorous mathematical framework.

Representation Theory: Algebras and Homomorphisms

Covers the goals and motivations of representation theory, focusing on associative algebras and homomorphisms.

Lie Algebra: Group Theory

Explores Lie Algebra's connection to Group Theory through associative operations and Jacobi identities.

Mock Exam: Solutions to the open question

Explores solutions to an open question from a mock exam, focusing on series convergence and function properties.

Real Sequences: Definitions and Operations

Explores real sequences, limits, and algebraic operations on sequences.

Fundamental Theorem of Algebra

Covers the Fundamental Theorem of Algebra, complex sequences, convergence, and continuity of functions.

Group Algebra: Maschke's Theorem

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Analytical Study of Space

Explores landmarks, coordinates, vectors, coplanarity, Cartesian equations, and geometric rules in space.