# Infinitesimal Calculus II (MAY211)

### General

School School of Science Department of Mathematics Undergraduate MAY211 2 Infinitesimal Calculus II Lectures, laboratory exercises (Weekly Teaching Hours: 5, Credits: 7.5) General Background None (from the typical point of view). Without the knowledge earned from the course “Infinitesimal Calculus I” will be nearly impossible to follow this course. Greek Yes (exams in English are provided for foreign students) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes This course is the sequel of the course “Infinitesimal Calculus I”. The student will get in contact with more notions and techniques in the branch of Analysis. In this course the students: Are taught the notions of convergence and absolute convergence of series. They learn criteria and theorems concerning these notions as well as they learn how to compute sums of series. They are introduced in the notion of power series and they learn how to calculate the radius of convergence of a power series. Are taught the notion of uniform continuity and they learn to distinguish this notion from continuity. Are taught the notion of Riemann integral and various theorems concerning this notion. They also learn various integrating techniques. Are taught Taylor’s theorem and they learn to write a given function as a Taylor series. The course provides inductive and analytical thinking, the students evolve their computational skills and they get knowledge necessary for other courses during their undergraduate studies.

### Syllabus

Series, convergence of series and criteria for convergence of series. Dirichlet’s criterion, D’ Alembert’s criterion, Cauchy’s criterion, integral criterion. Series with alternating signs and Leibnitz’s theorem. Absolute convergence and reordering of series, Power series, radius of convergence of power series.
Uniform continuity, definition and properties. Characterization of uniform continuity via sequences. Uniform continuity of continuous functions defined on closed intervals.
Riemann integral, definition for bounded functions defined on closed intervals. Riemann’s criterion, integrability of continuous functions. Indefinite integral and the Fundamental theorem of Calculus. Mean Value theorem of integral calculus, integration by parts, integration by substitution. Integrals of basic functions, integrations of rational functions. Applications of integrals, generalized integrals, relation between generalized integrals and series.
Taylor polynomials, Taylor’s Theorem, forms of the Taylor remainder. Taylor series and expansions of some basic functions as Taylor series.

### Teaching and Learning Methods - Evaluation

Delivery

Due to the theoretical nature of this course the teaching is exclusively given in the blackboard by the teacher.

Use of Information and Communications Technology

The students may contact their teachers by electronic means, i.e. by e-mail.

Teaching Methods