Chapter 10

प्रस्तावना

The preface of the book 'Calculus and Differential Equations' (कलन एवं अवकल समीकरण) introduces the structure, purpose, and approach of the text. The book is designed as per the curriculum prescribed by Vardhman Mahaveer Open University, Kota, for the B.Sc. Part I Mathematics, Paper II. The language is kept simple, engaging, and accessible, with careful use of English mathematical terminology alongside Hindi explanations to facilitate understanding. The book is authored by a team of experienced mathematicians and educators, ensuring factual accuracy and alignment with standard mathematical texts. The preface acknowledges the use of standard references and expresses gratitude to their authors. The book aims to help students prepare for competitive examinations as well as university assessments, providing clear guidance and comprehensive coverage of the subject matter.

• Book is structured as per the B.Sc. Part I Mathematics syllabus of Vardhman Mahaveer Open University, Kota.
• Language is simple and uses both Hindi and English mathematical terms.
• Content is prepared by a team of experienced mathematicians.
• Standard references have been used for accuracy.
• Book is useful for both university exams and competitive examinations.
• 📌 Calculus: The branch of mathematics that deals with the study of change (differentiation) and accumulation (integration).
• 📌 Differential Equations: Equations involving derivatives of functions, representing rates of change.

इकाई 1: श्रेणी (Series)

Unit 1 introduces the concept of series in mathematics. The unit outline lists all the topics to be covered, starting from the definition and types of series, partial sums, convergence and divergence, important properties, necessary conditions for convergence, geometric series, comparison tests, hyper-harmonic series, and various convergence tests (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, vocabulary, answers to self-assessment questions, and practice questions. The objectives clearly state that after studying this unit, students will be able to understand the types of series, distinguish between convergent, divergent, and oscillatory series, and comprehend the difference between absolutely and conditionally convergent series. The unit also aims to clarify the impact of adding, removing, or multiplying terms of an infinite series by a nonzero constant on its convergence or divergence.

• Unit covers all fundamental aspects of series.
• Students will learn to distinguish between different types of series.
• Various convergence tests are introduced.
• Absolute and conditional convergence are explained.
• Includes self-assessment and practice questions.
• 📌 Series: The sum of the terms of a sequence.
• 📌 Convergent Series: A series whose partial sums approach a finite limit.
• 📌 Divergent Series: A series whose partial sums do not approach a finite limit.