# Serlo: EN: Real Analysis

* ↳ Project "Serlo"
* ↳ Real Analysis Contents "Real Analysis" Help Introduction  Complex numbers  Supremum and infimum  Sequences  Convergence and divergence  Subsequences, Accumulation points and Cauchy sequences  Series  Convergence criteria for series  Exponential and Logarithm functions  Trigonometric and Hyperbolic functions  Continuity  Differential Calculus
We believe that anyone can understand higher level maths - with the right explanation! That's why we - a group of passionate students and teachers of mathematics - are translating our legendary co-created textbook for real analysis (“Analysis 1”) from German into English. If you know German and want to help us to make higher level maths accessible to everyone, then we look forward to hearing from you at en@serlo.org.

## Help

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## Introduction

* What is analysis?
* Why study analysis?
* Propositional logic
* Mathematical induction
* Real numbers

## 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

* Sequences
* Explicit and recursive description
* Examples and properties of sequences
* Exercises

## 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

* Sine and cosine

## 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
We believe that anyone can understand higher level maths - with the right explanation! That's why we - a group of passionate students and teachers of mathematics - are translating our legendary co-created textbook for real analysis (“Analysis 1”) from German into English. If you know German and want to help us to make higher level maths accessible to everyone, then we look forward to hearing from you at en@serlo.org.
Feedback? Do you want to join?
If you have questions concerning the content, or didn't understand something, the feel free to contact us! We would love to answer your questions! Also we are thankful for critics and/or comments! If you share our vision to explain university math in an comprehensible way, then contact us under:
E-Mail: en@serlo.org
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