# Calculus-2 2023/24

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## Teachers and Assistants

Group 221 (M+P+) 222 223 224
Lecturer Andrey Mazhuga
Teacher Andrey Mazhuga Корней Томащук (tg) Корней Томащук (tg) Павел Жуков (tg)
Consultations Sat, 17:00 -- 21:00, via Zoom
One must notify me beforehand
Assistant Алмасбек Стамбеков
(tg)
Елизавета Аникина
(tg)
Елизавета Аникина
(tg)
Тимур Нигматзянов
(tg)

## Course Description

This page contains basic information for the course Calculus-2 in 2023/2024 academic year at Bachelor’s Programme 'HSE and University of London Double Degree Programme in Data Science and Business Analytics' (DSBA).

During the course, the student will be formally graded on the following:

• one in-class oral test (O);
• one in-class written test (= midterm test) (W);
• several quizzes (Q, where Q is the average grade of all the quizzes in the course);
• several homework assignments (H, where H is the average grade of all the homework assignments in the course);
• one written exam (E).

All grades (namely, O, W, Q, H, and E) are real numbers from 0 to 10.

The cumulative grade, C, is obtained without rounding by the following formula:

C = 4/17*H + 4/17*Q + 4/17*W + 5/17*O.

The final grade for the course, F, is obtained by the following formula:

F = Round(7/10*C + 3/10*E).

where the function Round(x) is defined as follows: if the decimal part of x is less than 0.2, the grade is rounded downwards; if the decimal part of x is greater than 0.7, the grade is rounded upwards; if the decimal part of x is from the interval [0.2;0.7] and the student's seminar attendance during the first semester is not below 66%, the grade is rounded upwards; otherwise the grade is rounded downwards.

## Lecture Notes

If you notice an error or do not understand something, please describe it in the Bugs Table.

Module 2

• Lecture 8 (01.11.2023) Analytic functions; differentiability classes; smooth functions; the Taylor series; the Taylor formula; criteria for analyticity; the Borel lemma; some pathological examples of smooth functions.

Module 1

• Lecture 7 (18.10.2023) Power series; the radius and the interval of convergence of a power series; the Cauchy-Hadamard Formula; theorem on the uniform convergence of power series; theorem on derivative of power series; theorem on analyticity of power series; theorem on antiderivative of power series; theorem on equality of power series; the Abel theorem on the Cauchy product (proof).
• Lecture 6 (11.10.2023) Theorem on changing the order of two limits; theorem on continuity of a limit function; theorem on Riemann integrability of a limit function; theorem on integration of a uniformly convergent sequence; theorem on differentiability of a limit function; theorem on continuity of a series; theorem on term-by-term integration of a uniformly convergent series; theorem on term-by-term differentiation of a series; the Stone-Weierstrass theorem (without a proof).
• Lecture 5 (04.10.2023) Functional sequences and series; poinwise convergence; uniform convergence; the Cauchy criterion for the uniform convegrence; the negation of the Cauchy criterion for the uniform convegrence; the nevessary condition for the uniform convergence; the Weierstrass M-test; the Dirichlet test for the uniform convergence; the Abel test for the uniform convergence.
• Lecture 4 (27.09.2023) Product of series; the Mertens theorem; the Abel theorem on the Cauchy product; the Abel theorem on series products; the Wallis formula; the Stirling formula; the Robbins formula.
• Lecture 3 (20.09.2023) The Abel transformation; the Dirichlet test; the Abel test; the Leibniz test; sine and cosine sums; absolutely and conditionally convergent series; alternating p-series; the Cauchy rearrangement theorem; the Riemann rearrangement theorem.
• Lecture 2 (13.09.2023) Series with non-negative terms; the Cauchy Condensation test; the first comparison test; the second comparison test; p-series convergence; the integral test; the root test; the ration test; the Kummer test; the Bertrand test; the Gauss test.
• Lecture 1 (06.09.2023) Series; convergence and divergence; Cauchy's criterion and its negation; necessary condidion for convergence; the tale (= remainder) of a series; linear combination of series; grouping theorem.

Module 2

Module 1

Module 2

Module 1

## Homework

### The homework for group 221:

Module 2

• HW 13 (release: 10.12.23; deadline: 17.12.23 [23:59 MSK])
• no HW 12 (since Oral Test)
• HW 11 (release: 27.11.23; deadline: 03.12.23 [23:59 MSK])
• HW 10 (release: 19.11.23; deadline: 26.11.23 [23:59 MSK]) (seminar notes)
• HW 9 (release: 12.11.23; deadline: 19.11.23 [23:59 MSK])
• no HW 8(since Midterm)

Module 1

• HW 7 (release: 24.10.23; deadline: 05.11.23 [23:59 MSK])
• no HW 6(since we had two seminars in one day)
• HW 5 (release: 14.10.23; deadline: 22.10.23 [23:59 MSK])
• HW 4 (release: 07.10.23; deadline: 15.10.23 [23:59 MSK])
• HW 3 (release: 29.09.23; deadline: 08.10.23 [23:59 MSK])
• HW 2 (release: 22.09.23; deadline: 01.10.23 [23:59 MSK])
• HW 1 (release: 15.09.23; deadline: 24.09.23 [23:59 MSK])

### The homework for groups 222, 223:

Module 2

• HW 10 (release: 26.11.23; deadline: 03.12.23)
• HW 9 (release: 19.11.23; deadline: 26.11.23)
• HW 8 (release: 14.11.23; deadline: 19.11.23)
• HW 7 (release: 05.11.23; deadline: 12.11.23)

Module 1

• HW 6 (release: 22.10.23; deadline: 05.11.23)
• HW 5 (release: 15.10.23; deadline: 22.10.23)
• HW 4 (release: 08.10.23; deadline: 15.10.23)
• HW 3 (release: 01.10.23; deadline: 08.10.23)
• HW 2 (release: 23.09.23; deadline: 01.10.23)
• HW 1 (release: 16.09.23; deadline: 24.09.23)

### The homework for group 224:

Module 1

• [??? HW 1] (release: ???; deadline: ???) ([??? seminar notes])

221 222 223 224