Calculus 2, part 2 of 2: Sequences and series
What you'll learn
- How to solve problems concerning sequences and series (illustrated with 378 solved problems) and why these methods work.
- Various ways of defining number sequences: with help of a descriptive definition, an explicit (closed) formula, a recursive definition.
- Limits of number sequences, both finite and infinite (repetition and continuation of the topic introduced in Calc1p1).
- Arithmetic on extended reals (repetition and continuation of the topic introduced in Calc1p1).
- Dealing with indeterminate forms (repetition and continuation of the topic introduced in Calc1p1).
- Standard limits, comparing infinities (repetition and continuation of the topic introduced in Calc1p1).
- Squeeze Theorem for sequences (repetition and continuation of the topic introduced in Calc1p1).
- Weierstrass' Theorem for sequences (repetition and continuation of the topic introduced in Calc1p1).
- Functions defined for all positive arguments and their role in examining number sequences (monotonicity, limits).
- Solving recurrence relations: closed formula for linear recurrences of order two (with or without initial conditions).
- Stolz-Cesàro Theorem for computing limits of indeterminate forms.
- Ratio test for sequences for determining convergence or divergence.
- Cauchy Theorem (convergence of the arithmetic means of a convergent sequence).
- AM-GM inequality; convergence of the geometric means of a convergent sequence.
- Examining the differences of consecutive terms of the sequence (a_n) for determining convergence of a_n/n.
- Examining the quotients of consecutive terms of the sequence (a_n) for determining convergence of the n-th roots of a_n.
- Some applications of sequences (discretisation, approximations); sequences in other courses.
- Cauchy sequences and completeness of the set of real numbers; Bolzano-Weirestrass' Theorem.
- Series as limits of partial sums of sequences: definition and examples.
- Convergent and divergent series: simple and advanced examples.
- Arithmetic and geometric series and their convergence.
- p-series, their relation to p-integrals; discussion of their convergence.
- Comparison criteria for series.
- Integral test for series.
- Ratio test for series.
- Root test for series.
- Sequences of real-valued functions (a very brief introduction).
- Series of real-valued functions (a very brief introduction).
- Power series and their radius of convergence.
- Taylor polynomials and Taylor series, smooth functions.
- Applications of Taylor polynomials for approximating values of functions.
- Applications of Taylor polynomials for computing limits of indeterminate forms.
- Sneak peek into the content of the next course (Real Analysis: metric spaces).
Requirements
- Calculus 1: Limits and continuity (or equivalent)
- Calculus 1: Derivatives with applications (or equivalent)
- Calculus 2: Integrals with applications (or equilvalent)
- You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.
Description
Calculus 2, part 2 of 2: Sequences and series
Single variable calculus
S1. Introduction to the course
You will learn: about the content of this course; you will also get a list of videos form our previous courses where the current topics (sequences and series) were discussed.
S2. Number sequences: a continuation from Calc1p1
You will learn: more about sequences, after the introduction given in Calc1p1 (Section 5): in this section we repeat some basic facts from Calc1p1: the concept of a sequence and its limit, basic rules for computing limits of both determinate and indeterminate forms; these concepts are recalled, and you also get more examples of solved problems.
S3. Weierstrass' Theorem: a continuation from Calc1p1
You will learn: here we continue (after Calc1p1) discussing monotone sequences and their convergence; the main tool is Weierstrass' Theorem, also called "Monotone Convergence Theorem"; after repetition of some basic facts, you will get a lot of solved problems that illustrate the issue in depth.
S4. Using functions while working with sequences
You will learn: in this section we move to the new stuff: a functional approach to sequences, that we weren't able to study in Calc1p1, as the section about sequences came before the section about functions (in the context of limits and continuity); how to use (for sequences) the theory developed for functions (derivatives, l'Hôpital's rule, etc).
S5. New theorems and tests for convergence of sequences
You will learn: various tests helping us computing limits of sequences is some cases: Stolz-Cesàro Theorem with some corollaries, the ratio test for sequences; we will prove the theorems, discuss their content, and apply them on various examples.
S6. Solving recurrence relations
You will learn: solving linear recursions of order 2 (an introduction; more will be covered in Discrete Mathematics).
S7. Applications of sequences and some more problems to solve
You will learn: various applications of sequences; more types of sequence-related problems that we haven't seen before (some problems here are really hard).
S8. Cauchy sequences and the set of real numbers
You will learn: more (than in Calc1p1) about the relationships between monotonicity, boundedness, and convergence of number sequences; subsequences and their limits; limit superior and limit inferior (reading material only: Section 3.6 on pages 50-55 in the UC Davis notes); Bolzano-Weierstrass Theorem; fundamental sequences (sequences with Cauchy property), their boundedness and convergence; construction of the set of real numbers with help of equivalence classes of fundamental sequences of rational numbers; the definition of complete metric spaces.
S9. Number series: a general introduction
You will learn: about series: their definition and interpretation, many examples of convergent and divergent series (geometric series, arithmetic series, p-series, telescoping series, alternating series); you will also learn how to determine the sum of series in some cases; we will later use these series for determining convergence or divergence of other series, that are harder to deal with.
S10. Number series: plenty of tests, even more exercises
You will learn: plenty of tests for convergence of number series (why they work and how to apply them): comparison tests, limit comparison test, ratio test (d'Alembert test), root test (Cauchy test), integral test.
S11. Various operations on series
You will learn: how the regular computational rules like commutativity and associativity work for series; Cauchy product of series; remainders, their various shapes and their role in approximating the sum of a series.
S12. Sequences of functions (a very brief introduction)
You will learn: you will get a very brief introduction to the topic of sequences of functions; more will be covered in "Real Analysis: Metric spaces"; the concepts of point-wise convergence and uniform convergence are briefly introduced and illustrated with one example each; these concepts will be further developed in "Real Analysis: Metric spaces".
S13. Infinite series of functions (a very brief introduction)
You will learn: you get a very brief introduction to the topic of series of functions: just enough to introduce the topic of power series in the next section.
S14. Power series and their properties
You will learn: the concept of a power series and different ways of thinking about this topic; radius of convergence; arithmetic operations on power series (addition, subtraction, scaling, multiplication); some words about differentiation and integration of power series term after term (optional).
S15. Taylor series and related topics: a continuation from Calc1p2
You will learn: (a continuation from Section 10 in "Calculus 1, part 2 of 2: Derivatives with applications") Taylor- and Maclaurin polynomials (and series) of smooth functions; applications to computing limits of indeterminate expressions and to approximating stuff.
Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.
A detailed description of the content of the course, with all the 272 videos and their titles, and with the texts of all the 378 problems solved during this course, is presented in the resource file
“001 List_of_all_Videos_and_Problems_Calculus_2_p2.pdf”
under video 1 ("Introduction to the course"). This content is also presented in video 1.
Who this course is for:
- University and college students wanting to learn Single Variable Calculus (or Real Analysis)
- High school students curious about university mathematics; the course is intended for purchase by adults for these students
Instructors
- 4.9 Instructor Rating
- 3,907 Reviews
- 31,101 Students
- 16 Courses
I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.
I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.
I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Toruń (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Toruń, which gave me a very solid foundation for everything else I have learned and taught later.
In my courses I teach various branches of university mathematics that I have learned from absolutely excellent lectures of my dear professors from Toruń: Mirosław Uscki (b.1946), Zbigniew Bobiński (b.1940), Paweł Jarek (1933-2013), and Stanisław Balcerzyk (1932-2005).
My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".
In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).
I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.
- 4.9 Instructor Rating
- 3,907 Reviews
- 31,101 Students
- 16 Courses
I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.