### Description

Course is part of bachelor degree program in mathematical sciences and therein basic studies. It is third in a series of online Calculus courses. The collection of the three courses

- MAT11006: Calculus 1A: Limits and Differentiation
- MAT11007: Calculus 1B: Integration
- MAT11008: Advanced Calculus

correspond to the collection of the three Finnish analysis courses

- MAT11003 Raja-arvot
- MAT11004 Differentiaalilaskenta
- MAT11005 Integraalilaskenta

However, no single course in either collection can substitute a course from the other.

The courses MAT11006: Calculus 1A: Limits and Differentiation and MAT11007: Calculus 1B: Integration are prerequisite for taking this course.

Online English course MAT21002 Series, or its Finnish equivalent "Sarjat" continues the theme of the calculus courses.

This is proof based calculus or real analysis course. Concepts familiar from earlier Calculus courses, such as continuity are defined precisely. Axioms of the real number system, especially the completeness of the ordering of the real numbers are studied.

Students will learn to prove statements about sets of real numbers or real valued functions based on exact definitions, with various proof techniques, such as estimating quantities with triangle inequality, so called epsilon-delta method, using induction, and proof by contradiction.

This course can be taken after the courses MAT11006 Calculus 1A: Limits and Differentiation and MAT11007 Calculus 1B: Integration.

This course is organized in the first period in the spring term.

Course covers the following main topics

- Axioms of real numbers
- Completeness, supremum and infimum
- Convergence of sequences
- Proofs of Bolzano-Weirstrass theorem, extreme value theorem, intermediate value theorem among others
- Continuity rigorously
- Uniform continuity
- Pointwise and uniform convergence of series of functions

All necessary study material can be found in the course area. An online free book, Trench: Elementary Real Analysis will be also used.

Studying in this online course follows a weekly cycle. Each week students are expected to:

- Watch the prerecorded lectures or read the text allocated for the next workshop at the class Moodle site.
- Submit solutions to workshop problems by Wednesday evening.
- Grade and give feedback to other studentsâ€™ workshop submissions by Sunday evening.
- Also ask questions, hints for solving problems etc. in the discussion forum.

Continuous formative assessment takes place throughout the course by weekly workshops. These will give homework extra credit maximum of 5 points.

Final paper and pencil exam will measure comprehensively the learning goals as specified in section 9.

Scale 1-5 of grades will be used. Passing course with grade 1 will require approximately half of the points of the exam and for the best grade approximately 5/6 of the exam points are required. Homework extra credit points will be added to the exam points.

This is an online course. All course material and activities can be found on the online course area. There is a final proctored paper and pencil exam at the end of the course.