Chapter 6

प्रस्तावना

The preface introduces the textbook 'Calculus and Differential Equations' prepared by Vardhman Mahaveer Open University, Kota, for the B.Sc. Part I Mathematics (Paper II) curriculum. The book aims to present mathematical concepts in a simple, interesting, and accessible language, with appropriate use of English terminology alongside Hindi for clarity. The content is structured to facilitate both conceptual understanding and preparation for competitive examinations. The preface acknowledges the contribution of various scholars and standard references used in the creation of the book. The book is divided into multiple units, each authored by subject experts, ensuring factual accuracy and comprehensive coverage of the syllabus.

• The book is designed for B.Sc. Part I Mathematics (Paper II) students.
• Language is kept simple and includes English terminology for mathematical clarity.
• Content is structured to help in both academic and competitive exam preparation.
• Multiple units cover a wide range of calculus and differential equations topics.
• 📌 Calculus: The branch of mathematics dealing with limits, functions, derivatives, integrals, and infinite series.
• 📌 Differential Equations: Equations involving derivatives of functions.

इकाई 1: श्रेणी (Series) - इकाई की रूपरेखा

This section outlines the structure of Unit 1, which is dedicated to the concept of Series. It lists all the subtopics that will be covered, providing a roadmap for the unit. These include the definition and types of series, sequence of partial sums, convergence and divergence of series, important properties, necessary conditions for convergence, convergence of geometric series, comparison tests, convergence of hyper-harmonic series, various tests for convergence and divergence of series with positive terms (including Cauchy's root test, D'Alembert's ratio test, logarithmic ratio test, Raabe's test, De Morgan and Bertrand's test, Cauchy's condensation test, Gauss's test), alternating series, Leibniz's test for alternating series, absolute and conditional convergence, summary, glossary, answers to self-assessment questions, and practice exercises.

• Unit 1 covers all fundamental aspects of series in calculus.
• Includes definitions, types, and properties of series.
• Explains convergence and divergence with necessary conditions.
• Details various tests for convergence, including for alternating and positive term series.
• Provides summary, glossary, and exercises for practice.
• 📌 Series: The sum of the terms of a sequence.
• 📌 Convergence: When the sum of an infinite series approaches a finite limit.
• 📌 Divergence: When the sum of an infinite series does not approach a finite limit.