Series and Multivariable Functions (MAT201) | GSÜ

Content

Objective

To teach the notion of convergence of sequences and series
To teach certain technics for testing their convergence
To enable participants to work with multivariable functions, their limits and derivatives

Course Content

Convergence of series and power series
Convergence tests
Taylor Series
Multivariate functions and their graphs
Their limits
Notion of continuity for multivariate functions
Partial and directional derivatives of multivariate functions
Differentiability of multivariate functions

Course Learning Outcomes

1.Treatment of convergence of series
2.Treatment of convergence of power series
3. Computation of Taylor Series
4. Graphing multivariate functions
5. Notions of limit and continuity of multivariate functions
6. Notions of partial and directional derivatives of multivariate functions
7. Differentiability of multivariate functions

Teaching and Learning Methods

Exercises, discussions and worksheets

References

[1] P. Liret et D. Martinais, Analyse, 2e année : cours et exercices avec solutions, Dunod.
[2] A. Alves de Sá et B. Louro, Sequences and Series : Theory and Practice, Springer, 2024.
[3] F. Q. Gouvêa, A Short Book on Long Sums : Infinite Series for Calculus Students, Springer, 2023.
[4] A. Moreau, Suites et séries de fonctions, Université de Lille, Licence de Mathématiques, 2e année.
[5] Calculus, James Stewart

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Series, Absolute convergence.
2	Series with positive terms. Comparison theorems. Riemann series.
3	Convergence tests: Cauchy, d'Alembert, Abel.
4	Alternating series
5	Power Series
6	Taylor Series
7	Midterm
8	Series of functions, pointwise and uniform convergence of series of functions.
9	Stone-Weierstrass theorem
10	Multivariate functions, their graphs
11	Limits of multivariate functions, continutiy
12	Partial derivatives, differentiability
13	Second derivatives, Schwarz theorem
14	Optimization of multivariate functions

Practice Topics
Week	Weekly Contents
1	Review on functions with one variable, their limits, continuity, differentiability, integrals
2	Series, Absolute convergence.
3	Series with positive terms. Comparison theorems. Riemann series.
4	Convergence tests: Cauchy, d'Alembert, Abel.
5	Alternating series
6	Power Series
7	Taylor Series
8	Solution of midterm
9	Series of functions, pointwise and uniform convergence of series of functions.
10	Stone-Weierstrass theorem
11	Multivariate functions, their graphs
12	Limits of multivariate functions, continutiy
13	Partial derivatives, differentiability
14	Second derivatives, Schwarz theorem

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	3	55
Contribution of final exam to overall grade	1	45
Total	4	100

In-Term Studies
Activities	Number	Contribution
Assignments	13	5
Presentation	0	0
Midterm Examinations (including preparation)	1	30
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	2	20
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	16	55

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways;				X
2	can properly state and use concepts and results of major mathematical interest;				X
3	masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools;				X
4	is able to express one’s mathematical ideas in an organised way both in written and oral forms;				X
5	understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...);				X
6	has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems;			X
7	has a theoretical and practical knowledge in computer science well adapted for learning a programming language;		X
8	has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions;	X
9	has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement;			X
10	masters French language as well as other foreign languages, to a level sufficient to study or work abroad.				X

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	5	70
Working Hours out of Class	14	3	42
Assignments	13	3	39
Presentation	0	0	0
Midterm Examinations (including preparation)	1	8	8
Project	0	0	0
Laboratory	0	0	0
Other Applications	2	6	12
Final Examinations (including preparation)	1	12	12
Quiz	2	8	16
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	1	6	6
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			205
Total Workload / 25			8.20
Credits ECTS			8