This book gathers together a novel collection of problems in mathematical analysis that are challenging and worth studying. They cover most of the classical topics of a course in mathematical analysis, and include challenges presented with an increasing level of difficulty. Problems are designed to encourage creativity, and some of them were especially crafted to lead to open problems which might be of interest for students seeking motivation to get a start in research.
The sets of problems are comprised in Part I. The exercises are arranged on topics, many of them being preceded by supporting theory. Content starts with limits, series of real numbers and power series, extending to derivatives and their applications, partial derivatives and implicit functions. Difficult problems have been structured in parts, helping the reader to find a solution. Challenges and open problems are scattered throughout the text, being an invitation to discover new original methods for proving known results and establishing new ones. The final two chapters offer ambitious readers splendid problems and two new proofs of a famous quadratic series involving harmonic numbers. In Part II, the reader will find solutions to the proposed exercises.
Undergraduate students in mathematics, physics and engineering, seeking to strengthen their skills in analysis, will most benefit from this work, along with instructors involved in math contests, individuals who want to enrich and test their knowledge in analysis, and anyone willing to explore the standard topics of mathematical analysis in ways that aren’t commonly seen in regular textbooks.
Author(s): Alina Sîntămărian, Ovidiu Furdui
Series: Problem Books in Mathematics
Edition: 1
Publisher: Springer
Year: 2021
Language: English
Pages: 556
Tags: limits; series; real numbers; power series; derivatives; partial derivatives; implicit functions; quadratic series; harmonic numbers;
Preface
Contents
Notations
Part I Theory and Problems
1 Sequences of Real Numbers
1.1 Limits of Sequences
1.2 Applications of Stolz–Cesàro Theorem, the ∞/∞ and the 0/0 Cases
1.3 Wolstenholme Sequences
1.4 Limits of Integrals
2 Series of Real Numbers
2.1 Miscellaneous Series
2.2 Applications of Abel's Summation Formula
2.3 Series with Positive Terms
2.4 Alternating Series
2.5 Series with Harmonic Numbers and Factorials
2.6 A Mosaic of Series
3 Power Series
3.1 Convergence and Sum of Power Series
3.2 Maclaurin Series of Elementary Functions
3.3 Gems with Numerical and Power Series
3.4 Single Zeta Series
3.5 Polylogarithm Series
3.6 Inequalities and Integrals
3.7 Generating Functions
3.8 Series with Harmonic and Skew-Harmonic Numbers
3.9 Remarkable Numerical and Function Series
3.10 Multiple Series with the Riemann Zeta Function
3.11 Series Involving Products of Harmonic Numbers
4 Derivatives and Applications
4.1 Apéritif
4.2 Integral Equations
4.3 Differential Equations
4.4 Higher Order Derivatives
4.5 Taylor's Formula
4.6 Series with the Maclaurin Remainder of a Function f
4.7 Series with Fractional Part Function
4.8 Extrema of One Variable Functions
5 Partial Derivatives and Applications
5.1 Partial Derivatives, the Jacobian and the Hessian Matrices, Differential Operators
5.2 The Chain Rule
5.3 Homogeneous Functions. Euler's Identity
5.4 Taylor's Formula for Real Functions of Two Real Variables
5.5 The Differential of Several Real Variable Functions
5.6 Extrema of Several Real Variable Functions
6 Implicit Functions
6.1 Implicit Functions of One Real Variable Defined by an Equation
6.2 Implicit Functions of Two Real Variables Defined by an Equation
6.3 Implicit Functions of One Real Variable Defined by a System of Equations
6.4 Implicit Functions of Two Real Variables Defined by a System of Equations
7 Challenges, Gems, and Mathematical Beauties
7.1 Limits of Sequences
7.2 Limits of Integrals
7.3 Convergence and Evaluation of Series
7.4 Harmonic Series
7.5 Series with Factorials
7.6 Series of Functions
7.7 Pearls of Series with Tails of Zeta Function Values
7.8 Exotic Zeta Series
7.9 Special Differential Equations
7.10 Inequalities
7.11 Fabulous Integrals
8 An Artistry of Quadratic Series: Two New Proofs of Sandham–Yeung Series
8.1 The First Proof
8.2 The Second Proof
Part II Solutions
9 Sequences of Real Numbers
9.1 Limits of Sequences
9.2 Applications of Stolz–Cesàro Theorem, the ∞/∞ and the 0/0 Cases
9.3 Wolstenholme Sequences
9.4 Limits of Integrals
10 Series of Real Numbers
10.1 Miscellaneous Series
10.2 Applications of Abel's Summation Formula
10.3 Series with Positive Terms
10.4 Alternating Series
10.5 Series with Harmonic Numbers and Factorials
10.6 A Mosaic of Series
11 Power Series
11.1 Convergence and Sum of Power Series
11.2 Maclaurin Series of Elementary Functions
11.3 Gems with Numerical and Power Series
11.4 Single Zeta Series
11.5 Polylogarithm Series
11.6 Inequalities and Integrals
11.7 Generating Functions
11.8 Series with Harmonic and Skew-Harmonic Numbers
11.9 Remarkable Numerical and Function Series
11.10 Multiple Series with the Riemann Zeta Function
11.11 Series Involving Products of Harmonic Numbers
12 Derivatives and Applications
12.1 Apéritif
12.2 Integral Equations
12.3 Differential Equations
12.4 Higher Order Derivatives
12.5 Taylor's Formula
12.6 Series with the Maclaurin Remainder of a Function f
12.7 Series with Fractional Part Function
12.8 Extrema of One Variable Functions
13 Partial Derivatives and Applications
13.1 Partial Derivatives, the Jacobian and the Hessian Matrices, Differential Operators
13.2 The Chain Rule
13.3 Homogeneous Functions. Euler's Identity
13.4 Taylor's Formula for Real Functions of Two Real Variables
13.5 The Differential of Several Real Variable Functions
13.6 Extrema of Several Real Variable Functions
14 Implicit Functions
14.1 Implicit Functions of One Real Variable Defined by an Equation
14.2 Implicit Functions of Two Real Variables Defined by an Equation
14.3 Implicit Functions of One Real Variable Defined by a System of Equations
14.4 Implicit Functions of Two Real Variables Defined by a System of Equations
15 Challenges, Gems, and Mathematical Beauties
15.1 Limits of Sequences
15.2 Limits of Integrals
15.3 Convergence and Evaluation of Series
15.4 Harmonic Series
15.5 Series with Factorials
15.6 Series of Functions
15.7 Pearls of Series with Tails of Zeta Function Values
15.8 Exotic Zeta Series
15.9 Special Differential Equations
15.10 Inequalities
15.11 Fabulous Integrals
References
Index
