# MATA174 Introduction to mathematical analysis 4 (5 cr)

**Study level:**

Intermediate studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2020-2021, 2021-2022, 2022-2023

## Tweet text

The main topics of the study module are improper integral, number series, function series and sequences.

## Description

Improper integrals. Number series. Sequences and series of functions and their convergence. Power series. Taylor polynomials and Taylor series.

## Learning outcomes

The student

- knows the definition(s) of improper integral and knows how to study their convergence
- knows the definition for the convergence of a number series and is able to recognize the most common series
- is able to use the most common convergence tests
- knows the definitions for point wise and uniform convergence for function sequences and series, and is able to use them
- knows which properties are stable under uniform convergence
- knows how to apply Weierstrass' test to study uniform convergence
- is familiar with the definitions of power series and its radius convergence, and knows how to study the convergence of a power series
- knows what are Taylor polynomials and how they can be used to study limits
- is able to recognize the most common power series representations

## Description of prerequisites

Introduction to mathematical analysis 1-3

## Study materials

Lecture notes (in Finnish)

Fitzpatrick: Advanced Calculus, chap. 8-9. or Brannan: A first course in Mathematical Analysis, chap. 3 and 8.

## Literature

- P.M Fitzpatrick: Advanced Calculus (2nd ed); ISBN: 978-0-8218-4791-6
- D. Brannan: A first course in Mathematical Analysis; ISBN: 13 978-0-521-6842-8

## Completion methods

### Method 1

**Description:**

Final exam and exercises. The weekly exercises are handed in and graded. To attend the final exam, the students must receive enough bonus-points from the exercises (the exact amount is mentioned in the teaching schedule).

**Evaluation criteria:**

Final exam and weekly exercises.

**Time of teaching:**

Period 4

Select all marked parts

### Method 2

**Description:**

Lopputentti. Luento-opetukseen osallistuneiden harjoitustehtäväpisteitä ei huomioida.

**Evaluation criteria:**

Final exam.

Select all marked parts

**Parts of the completion methods**

x

### Teaching (5 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Language:**

Finnish

**Study methods:**

Lectures (in Finnish) 48 h, exercises

#### Teaching

##### 8/28–8/28/2020 Exam

##### 3/15–5/23/2021 Lectures

##### 5/26–5/26/2021 Exam

##### 6/9–6/9/2021 Exam

x

### Exam (5 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Language:**

English, Finnish

**Study materials:**

Fitzpatrick: Advanced Calculus, chap. 8-9. or Brannan: A first course in Mathematical Analysis, chap. 3 and 8.