### Instruction

Name Cr Method of study Time Location Organiser
Series 5 Cr Examinarium (electronic exam room) 23.1.2020 - 31.12.2020
Sarjat, 4. periodi 5 Cr Lecture Course 10.3.2020 - 30.4.2020
Series 5 Cr Online Examination 5.5.2020 - 5.5.2020
Name Cr Method of study Time Location Organiser
Series 5 Cr Course exam 22.10.2019 - 22.10.2019
Sarjat, 1 periodi 5 Cr Lecture Course 3.9.2019 - 18.10.2019
Series 5 Cr Course exam 6.5.2019 - 6.5.2019
Series 5 Cr Online Course 11.3.2019 - 5.5.2019
Series 5 Cr Examination 23.10.2018 - 23.10.2018
Sarjat, 1 periodi 5 Cr Lecture Course 4.9.2018 - 19.10.2018
Series 5 Cr Online Course 12.3.2018 - 6.5.2018
Series 5 Cr General Examination 13.12.2017 - 13.12.2017
Series 5 Cr Examinarium (electronic exam room) 24.11.2017 - 22.1.2020

### Target group

NB: This page concerns online course held in English, to see information about the Finnish lecture based equivalent course, please change language to Finnish.

Course is part of bachelor degree program in mathematical sciences and therein intermediate studies.

### Prerequisites

Either the collection of the three Calculus courses

• MAT11006: Calculus 1A: Limits and Differentiation
• MAT11007: Calculus 1B: Integration

or the collection of the three Finnish analysis courses

• MAT11003: Raja-arvot

Students need to be fluent with finding limits, derivatives and integrals as well as proving results based on definitions, especially producing epsilon and delta type proofs.

### Learning outcomes

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

### Timing

This online course takes place in period IV.

### Contents

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. Maclaurin and Taylor Polynomials and Series, Approximating Functions

### Activities and teaching methods in support of learning

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. Solve quiz problems. They will help to prepare for the workshops.
3. Submit solutions to workshop problems by Wednesday evening.
4. Grade and give feedback to other students’ workshop submissions by Sunday evening.
5. Also ask questions, hints for solving problems etc. in the discussion forum.

### Study materials

All necessary study material can be found 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.

### Assessment practices and criteria

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 with maximum of 30 points will measure comprehensively the learning goals.

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.

### Recommended optional studies

The Finnish lecture based version MAT21002 Sarjat, in period I is equivalent to this course.

### Relation to other study units

Tässä kurssipaketilla tarkoitetaan toisiinsa kiinteästi liittyviä kursseja, jotka eivät kuitenkaan muodosta opintokokonaisuutta.

Seuraavat kurssipaketit ovat päällekkäisiä. Tämä tarkoittaa sitä, että voit sijoittaa matematiikan opintoihisi vain yhden seuraavista paketeista. Huomaa lisäksi, että vaikka kurssipaketit ovat sisällöllisesti päällekkäisiä, yksittäiset kurssit eivät vastaa toisiaan.

• MAT11003 / 57116 Raja-arvot, 5 op
• MAT11004 / 57117 Differentiaalilaskenta, 5 op
• MAT11005 / 57118 Integraalilaskenta, 5 op
• (MAT21002 / 57119 Sarjat, 5 op)

• 57016 Analyysi I 10 op (Raja-arvot, Differentiaalilaskenta)
• 57017 Analyysi II 10 op (Integraalilaskenta, Sarjat)

• MAT11006 Calculus 1A: Limits and Differentiation
• MAT11007 Calculus 1B: Integration
• (MAT21002 Series, 5 op)

Vastaavat vanhat kurssit Calculus I 8 op, Calculus II 8 op ja Advanced Calculus 6 op. Vanha kurssi Analyysin peruskurssi 10 op vastaa karkeasti kursseja Calculus 1A ja Calculus 1B.

Kurssipaketti 3 – ei pääaineopiskelijoille (taloustieteen opiskelijoille)

• MAT11010 Matemaattinen analyysi I, 5 op
• MAT11011 Matemaattinen analyysi II, 5 op
• MAT11012 Matemaattinen analyysi III, 5 op
• MAT11013 Matemaattinen analyysi IV, 5 op