Chapter 8

प्रस्तावना

The Preface introduces the textbook 'Calculus and Differential Equations' published by Vardhman Mahaveer Open University, Kota. The book is designed according to the curriculum for B.Sc. Part I Mathematics, Paper II. The language of the book is intentionally kept simple, engaging, and accessible. A conscious effort has been made to use appropriate English terminology alongside Hindi to suit the nature of mathematics. Various units of the book are authored by distinguished scholars, who have referred to standard texts to ensure factual accuracy. The book aims to guide students preparing for competitive exams as well. The Preface emphasizes the importance of clarity in mathematical concepts and the use of both Hindi and English mathematical vocabulary. The authors express gratitude to the creators of standard references used in the preparation of the book. The book is intended to be helpful for both regular study and exam preparation, providing a clear pathway for students to understand and master calculus and differential equations.

• The book is tailored for B.Sc. Part I Mathematics, Paper II.
• Language is simple, engaging, and accessible.
• English mathematical terminology is used alongside Hindi.
• Units are written by expert mathematicians.
• Standard reference books were used for accuracy.
• The book is useful for competitive exam preparation.
• 📌 Calculus: The branch of mathematics dealing with limits, derivatives, integrals, and infinite series.
• 📌 Differential Equations: Mathematical equations involving derivatives which describe various phenomena.

इकाई 1: श्रेणी (Series)

Unit 1 focuses on the concept of Series, which is a fundamental topic in calculus. The outline of the unit includes objectives, introduction, definition and types of series, sequence of partial sums, convergence and divergence of series, important properties of infinite series, necessary conditions for convergence, geometric series, comparison tests, hyper-harmonic series, various tests for convergence and divergence of series with positive terms, alternating series, Leibniz's test for alternating series, absolute and conditional convergence, summary, glossary, answers to self-evaluation questions, and practice questions. The objectives of this unit are: - To understand the main types of series. - To distinguish between convergent, divergent, and oscillatory series. - To comprehend the difference between absolutely and conditionally convergent series. - To know the necessary restrictions for convergence of a series. - To realize that adding or subtracting a finite number of terms or multiplying each term by a non-zero constant does not affect the convergence or divergence of an infinite series. The introduction states that the unit will discuss infinite series, as the sum of a finite series is always definite and does not require further analysis.

• Unit 1 covers the concept of Series in calculus.
• Objectives include understanding types and properties of series.
• Distinguishes between convergent, divergent, and oscillatory series.
• Explains absolute and conditional convergence.
• Discusses necessary conditions for convergence.
• Adding/removing finite terms or multiplying by a constant does not affect convergence.
• 📌 Series: The sum of the terms of a sequence.
• 📌 Convergent Series: A series whose sum approaches a finite value.
• 📌 Divergent Series: A series whose sum does not approach a finite value.