Chapter 4

प्रस्तावना

The preface of the book 'Calculus and Differential Equations' (कलन एवं अवकल समीकरण) outlines the purpose and approach of the text. This book is specifically designed for the B.Sc. Part I Mathematics, Paper II, as per the syllabus prescribed by Vardhman Mahaveer Open University, Kota. The authors have made a concerted effort to ensure that the language of the book is simple, engaging, and easy to understand, keeping in mind the nature of mathematics as a subject. The book uses appropriate English terminology alongside Hindi to facilitate better comprehension. Each unit of the book has been authored by subject experts, and standard reference books have been consulted to ensure factual accuracy. The book is also intended to be helpful for students preparing for competitive examinations.

• The book is tailored for B.Sc. Part I Mathematics, Paper II.
• Language is kept simple and accessible, with English terminology included.
• Units are written by subject experts using standard references.
• Aims to assist students in both university exams and competitive exams.
• 📌 Calculus: The branch of mathematics dealing with derivatives and integrals.
• 📌 Differential Equations: Equations involving derivatives of functions.

इकाई 1: श्रेणी (Series)

Unit 1 focuses on the concept of Series (श्रृंखला), which is a fundamental topic in calculus and mathematical analysis. The unit outline provides a roadmap of the topics covered, including definitions, types of series, convergence and divergence, important properties, and various tests for convergence. The unit also covers alternating series, absolute and conditional convergence, and includes summaries, terminology, and exercises for practice. The objectives at the beginning clarify what students will learn, such as distinguishing between different types of series, understanding convergence and divergence, and applying various tests to determine the nature of a series.

• Covers definitions and types of series.
• Explains convergence, divergence, and oscillatory series.
• Includes important properties and necessary conditions for convergence.
• Describes various convergence tests (e.g., Cauchy's root test, D'Alembert's ratio test).
• Discusses alternating series and absolute convergence.
• Provides summaries, terminology, and exercises.
• 📌 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.