Analysis I covers Fundamentals of Mathematical Analysis: Convergence of Sequences and Series, Continuity, Differentiability, Riemann Integral, Sequences and Series of Functions, Uniformity, Interchange of Limit Operations. Three versions of the course are available. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. Option A chooses less abstract definitions and proofs, and gives applications where possible. Option B is more demanding and is for students with more mathematical maturity; it places more emphasis on Point-Set Topology and N-Space, whereas Option A is concerned primarily with the Real Line. Option C is a variant of Option B, with further instruction and practice in written and oral communication.

This free online textbook is a one semester course in basic analysis. These were my lecture notes for teaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in fall 2009. The course is a first course in mathematical analysis aimed at students who do not necessarily wish to continue a graduate study in mathematics. A Sample Darboux sums prerequisite for the course is a basic proof course. The course does not cover topics such as metric spaces, which a more advanced course would. It should be possible to use these notes for a beginning of a more advanced course, but further material should be added.

CK-12 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and problem solving for teaching CK-12 Calculus Student Edition. The solution guide is available upon request.

Love math but bored in math class? This is the course for you! Combinatorics is a fascinating branch of mathematics that applies to problems ranging from card games to quantum physics to the internet. The only pre-requisite is basic algebra; however we will be covering a lot of material. A mathematically agile mind will be helpful.

This module will present an introduction into convergence and focus on what a sequence is and how it behaves as it approaches infinity.

This book covers the following topics: Sequences, limits, and difference equations; functions and their properties; best affine approximations; integration; polynomial approximations and Taylor series; transcendental functions; the complex plane; differential equations.

This course describes discrete mathematics, which involves processes that consist of sequences of individual steps (as compared to calculus, which describes processes that change in a continuous manner). The principal topics presented in this course are logic and proof, induction and recursion, discrete probability, and finite state machines. Upon successful completion of this course, the student will be able to: Create compound statements, expressed in mathematical symbols or in English, to determine the truth or falseness of compound statements and to use the rules of inference to prove a conclusion statement from hypothesis statements by applying the rules of propositional and predicate calculus logic; Prove mathematical statements involving numbers by applying various proof methods, which are based on the rules of inference from logic; Prove the validity of sequences and series and the correctness or repeated processes by applying mathematical induction; Define and identify the terms, rules, and properties of set theory and use these as tools to support problem solving and reasoning in applications of logic, functions, number theory, sequences, counting, probability, trees and graphs, and automata; Calculate probabilities and apply counting rules; Solve recursive problems by applying knowledge of recursive sequences; Create graphs and trees to represent and help prove or disprove statements, make decisions or select from alternative choices to calculate probabilities, to document derivation steps, or to solve problems; Construct and analyze finite state automata, formal languages, and regular expressions. (Computer Science 202)

Signals can be represented by discrete quantities instead of as a function of a continuous variable. These discrete time signals do not necessarily have to take real number values. Many properties of continuous valued signals transfer almost directly to the discrete domain.

CK-12 Foundation's Math Analysis FlexBook is a rigorous text that takes students from analyzing functions to mathematical induction to an introduction to calculus.

This module represents points for the conductor to consider regarding rhythm in choral works. These include rhythmic structure, sequences, types, meter changes and identifying places in the music where the chorus will have difficulty with the rhythm.

An updated version of the Sequences module.

A teacher's guide for lecturing on arithmetic and geometric sequences.

The prerequisite subjects needed for sequences and series.

An updated version of the Sequences and Series module.

An updated version of the Homework: Arithmetic and Geometric Sequences module.

An updated version of the Homework: Arithmetic and Geometric Series module.

An updated version of the Homework: Proof by Induction module.

An updated version of the Homework: Series and Series Notation module.

An updated version of the Sample Test: Sequences and Series module.