Sequences and series of real numbers

By ENG SAID

Sequences and Series of Real Numbers 90 pages, 6 x 9 inches This comprehensive booklet provides an in-depth exploration of sequences and series of real numbers, covering fundamental concepts, theorems, and applications, It is an essential resource for students, researchers, and professionals seeking to understand and work with sequences and series. Sequences The booklet begins by defining sequences and exploring their properties, including: Definition: Sequence: A sequence is a function that assigns a real number to each positive integer. Range of Sequence: The set of all values that a sequence can take. Bounded Sequence: A sequence that is bounded above and below Bounded Below: A sequence that is bounded below by a real number Greatest Lower Bound or Indium: The largest lower bound of a sequence. Monotonic Sequence: A sequence that is either monotonically increasing or decreasing. Limit of Sequence: The limit of a sequence as it approaches infinity. Oscillatory Sequence: A sequence that oscillates between different values. Sandwich Theorem or Sequence Rule: A theorem that provides a way to determine the limit of a sequence. The booklet also covers Cauchy theorems on limits, observations of convergence squeeze, Cauchy's squeeze, and the monotonic convergence theorem. Subsequences and Limit Points The booklet explores subsequences and limit points, including: Observations of Subsequence: Properties of subsequences and their limits. Limit Point: A point that is a limit point of a sequence. Limit Superior and Limit Inferior: The largest and smallest limit points of a sequence. Observations of Limit Superior and Inferior: Properties of limit superior and inferior. Series The booklet then turns to series, covering: Series: A series is a sum of a sequence of real numbers. Sequence and Partial Sum of Series: The relationship between a sequence and its partial sums. Oscillatory Series: A series that oscillates between different values. Cauchy General Principal of Convergence: A general principle for determining the convergence of a series. Some General Principal of Infinite Series: General principles for working with infinite series. Tests for Convergence The booklet provides a range of tests for convergence, including: Test of Convergence of Infinite Series: Tests for determining the convergence of an infinite series. Comparison Theorem: A theorem that provides a way to compare series. Comparison Test: A test that uses comparison to determine convergence. Working Rule for Using Comparison Test: A guide to using the comparison test. Cauchy's Root Test: A test that uses the root of a series to determine convergence. Logarithmic Test: A test that uses logarithms to determine convergence. Cauchy's Condensation Test: A test that uses condensation to determine convergence. DE - Morgan and Bertrand Test: A test that uses De Morgan and Bertrand's theorem to determine convergence. Higher Logarithmic Test: A test that uses higher logarithms to determine convergence. Cauchy's Integral Test: A test that uses integration to determine convergence. Kumar's Test: A test that uses Kumar's theorem to determine convergence. Gauss's Test: A test that uses Gauss's theorem to determine convergence. Exercises: A range of exercises to help readers practice and apply the concepts. Answers: Solutions to the exercises. This booklet provides a comprehensive introduction to sequences and series of real numbers, covering fundamental concepts, theorems, and applications, It is an essential resource for students, researchers, and professionals seeking to understand and work with sequences and series.

• Format: Paperback
• Language: English
• ISBN: B0F66KZPJ1
• ISBN13: 9798281135764
• Release Date: April 2025
• Publisher: Independently published
• Length: 90 Pages
• Dimensions: 9.0" x 0.2" x 6.0"