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- 244 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android
eBook - ePub
Problems in Mathematical Analysis
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About This Book
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
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Information
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 [θcn] is prime for every positive integer n.
1.3 Show that there exists a real number θ such that each number of the form is prime.
1.4 Let (an) be an arbitrary sequence of integers greater than 1. Prove that every real number x ∈[0, 1) can be represented as , where xk ∈ {0, 1, …, ak − 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 , where (nk) is a sequence of positive integers such that nk+1/nk ∈ {2, 3, 4}.
1.6 Prove that if an ≠ 0, n = 1, 2, … and limn→∞ an = 0, then for every real number x there exists the integer sequences (kn), (mn) such that and .
1.7 Show that if for every natural n 0 < an < and , then for each x ∈(0, 1) there exists a subsequence (kp) satisfying .
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 (ck) such that
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