Description
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
 Sequences and Series
 Convergence
 Geometric Series, pseries
 Alternating Series
 Convergence Tests: Integral, Comparison, Ratio, Root, Absolute Convergence
 Power Series, Abel’s Theorem, Radius of Convergence
 Uniform convergence, Differentiating and Integrating Series
 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:
 Watch the prerecorded lectures or read the text allocated for the next workshop at the class Moodle site.
 Submit solutions to workshop tasks by Wednesday evening. Late submissions will not be accepted.
 Grade and give feedback to other students’ 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 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 15 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