Kaisa_2012_3_photo by Veikko Somerpuro

This online course can be completed by
- taking an Examinarium exam (if available)
- submitting an extensive homework essay amounting 10 pages of work about core topics of the course
- online oral examination
- other means are supported

Please see for more information course's Moodle area or ask Matti Pauna.

10.2.2020 at 09:00 - 3.5.2020 at 23:59


Programme: Bachelor's programme in Science. (the course is taught by Matemaattisten tieteiden kandiohjelma)

Module: Basics studies in Mathematics.

The course is optional within the programme but compulsory for the mathematics study track.

The course is available to the students of other programmes upon agreement with the teacher.

Calculus IA: Limits and Differentiation
Calculus IB: Integration
Advanced Calculus


After successfully completing this course students will

  • Be familiar with concepts of infinite sequences and series
  • Understand the principles of how a sum of infinite series is formed
  • Recognize different types of series
  • Understand what is meant by convergence and divergence of a sequence and series
  • Master various tests for convergence of series and discern situations in which to apply them
  • Represent elementary functions, such as trigonometric functions and logarithms, as series, find domains in which these series converge, and use these series representations to approximate values of functions
  • Understand concept of uniform convergence, determine whether a series converges uniformly, and use it to differentiate and integrate a series term by term

Within mathematics study track, first year of studies.

In other study tracks, according to schedule of track/programme

Course covers the following main topics

  1. Sequences and Series
  2. Convergence
  3. Geometric Series, p-series
  4. Alternating Series
  5. Convergence Tests: Integral, Comparison, Ratio, Root, Absolute Convergence
  6. Power Series, Abel’s Theorem, Radius of Convergence
  7. Uniform convergence, Differentiating and Integrating Series
  8. Maclaurin and Taylor Polynomials and Series, Approximating Functions

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.

All necessary study material can be found in the course area. Most topics are covered in videos created by prof. Mika Seppälä. Some topics are covered in the book "Introduction to Real Analysis" by Trench, available in the course area.

Standard Calculus text books, such as Adams' or Stewart's Calculus books can be used as side material among others.
The parts covering Series in the Finnish “Analyysia reaaliluvuilla” (Harjulehto, Klén, Koskenoja) can be recommended for Finnish speaking students (though language of communication is always English).
NB: sequence of covering topics in these books may vary from that in the course.

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

  1. Watch the prerecorded lectures or read the text allocated for the next workshop at the class Moodle site.
  2. Submit solutions to workshop tasks by Wednesday evening. Late submissions will not be accepted.
  3. Grade and give feedback to other students’ workshop submissions by Sunday evening.
  4. Also ask questions, hints for solving problems etc. in the discussion forum.

Continuous formative assessment takes place throughout the course by weekly quizzes and 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.

The course is offered in the spring every year in Period IV

Matti Pauna