Calculus 2, part 2 of 2: Sequences and series
Single variable Calculus: sequences and series of numbers and of real-valued functions of one real variable
Calculus 2, part 2 of 2: Sequences and series free download
Single variable Calculus: sequences and series of numbers and of real-valued functions of one real variable
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.
