Real Analysis – Serlo
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Introduction
Complex numbers
- Introduction and motivation
- Definition of complex numbers
- Absolute value and conjugation
- Polar representation
- Drawing complex-valued functions
- Exercises
Supremum and infimum
- Supremum and infimum
- The infinite case
- How to prove existence of a supremum or infimum
- Properties of supremum and infimum
Sequences
Convergence and divergence
- Definition of limit
- How to prove convergence and divergence
- Examples for limits
- Unbounded sequences diverge
- Limit theorems
- The squeeze theorem
- Monotony criterion
- How to prove convergence for recursive sequences
- Exercises
Subsequences, Accumulation points and Cauchy sequences
- Subsequence
- Accumulation points of sequences
- Accumulation points of sets
- The Bolzano-Weierstrass theorem
- Divergence to infinity
- Divergence to infinity: rules
- Lim sup and lim inf
- Cauchy sequences
- Exercises
Series
- Series
- Computation rules for series
- Telescoping sums and series
- Geometric series
- Harmonic series
- Exponential series
- Absolute convergence of a series
- Rearrangement theorem for series
- Exercises
Convergence criteria for series
- Overview: convergence criteria
- Cauchy criterion
- Term test
- Bounded series and convergence
- Direct comparison test
- Root test
- Ratio test
- Alternating series test
- Cauchy condensation test
- Application of convergence criteria
- Exercises
Exponential and Logarithm functions
- Derivation and definition of the exponential series
- Properties of the exponential series
- Logarithmic function
- Real exponents
- Exp and log functions for complex numbers
- Exercises
Trigonometric and Hyperbolic functions
Continuity
- Continuity of functions
- Epsilon-delta definition of continuity
- Sequential definition of continuity
- Limit of functions
- Proving continuity
- Proving discontinuity
- Composition of continuous functions
- Extreme value theorem
- Intermediate value theorem
- Continuity of the inverse function
- Uniform continuity
- Lipschitz continuity
- Exercises
Differential Calculus
- Derivatives
- Computing derivatives
- Computing derivatives - special
- Derivative - inverse function
- Examples for derivatives
- Derivatives of higher order
- Rolle's theorem
- Mean value theorem
- Constant functions
- Monotone functions
- Derivative and local extrema
- L'Hôspital's rule
- Overview: continuity and differentiability
- Exercises 1
- Exercises 2
- Exercises 3
- Exercises 4
