Problems in Mathematical Analysis

Biler

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9780824783129
• eBook ISBN: 9781351421454
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Chapter 1 poses 134 problems concerning real and complex numbers, chapter 2 poses 123 problems concerning sequences, and so it goes, until in chapter 9 one encounters 201 problems concerning functional analysis. The remainder of the book is given over to the presentation of hints, answers or referen

1

Real and Complex Numbers

Topics covered include:

prime numbers, representation of reals as series, group and topological structure of real numbers, polynomials, inequalities, rational and irrational numbers, symmetric functions, geometric properties of complex numbers, various problems.

1.1 Show that an irrational power of an irrational number can be rational.

1.2 Prove that if c > 8/3, then there exists a real number θ such that [θ^cn] is prime for every positive integer n.

1.3 Show that there exists a real number θ such that each number of the form $\left[\begin{array}{lll}\hfill & \hfill & 2^{\theta }\hfill \\ \hfill & ⋰\hfill & \hfill \\ 2^2\hfill & \hfill & \hfill \end{array}\right]$ is prime.

1.4 Let (a_n) be an arbitrary sequence of integers greater than 1. Prove that every real number x ∈[0, 1) can be represented as $x={\sum }_{k=1}^{\infty }{x}_k/a_1{a}_2\cdots a_k$, where x_k ∈ {0, 1, …, a_k − 1}. Give necessary and sufficient conditions for the existence of two such representations of the same number x.

1.5 Show that every x ∈(0, 1] can be represented as $x={\sum }_{k=1}^{\infty }1/n_k$, where (n_k) is a sequence of positive integers such that n_k+1/n_k ∈ {2, 3, 4}.

1.6 Prove that if a_n ≠ 0, n = 1, 2, … and lim_n→∞ a_n = 0, then for every real number x there exists the integer sequences (k_n), (m_n) such that $x={\sum }_{n=1}^{\infty }{k}_n{a}_n$ and $x={\displaystyle \prod {}_{n=1}^{\infty }}{m}_n{a}_n$.

1.7 Show that if for every natural n 0 < a_n < ${\sum }_{k=n+1}^{\infty }{a}_k$ and ${\sum }_{n=1}^{\infty }{a}_n=1$, then for each x ∈(0, 1) there exists a subsequence (k_p) satisfying ${\sum }_{p=1}^{\infty }{a}_{k_p}=x$.

1.8 Given a countable subset C of (0, 1) find necessary and sufficient conditions for the existence for every x ∈(0, 1) of a rearrangement of C in a sequence (c_k) such that ${\sum }_{k=1}^{\mathrm{\infty ...}}$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1. Real and complex numbers
• 2. Sequences
• 3. Series
• 4. Functions of one real variable
• 5. Functional equations and functions of several variables
• 6. Real analysis, measure and integration
• 7. Analytic functions
• 8. Fourier series
• 9. Functional analysis
• Answers
• References
• Index