Chapter 9

Table of Contents

This section presents the table of contents for the book 'Calculus and Differential Equations' as prescribed by Vardhman Mahaveer Open University, Kota. It outlines the structure of the book, listing all the units and their respective page numbers. The units cover a wide range of topics in calculus and differential equations, including Series, Generalised Mean Value Theorem and Power Series, Derivative of Length of a Curve and the Pedal Equation, Curvature, Partial Differentiation, Maxima and Minima, Asymptotes, Curve Tracing, Envelope, Area of Plane Curves (Quadrature), Length of Plane Curves (Rectification), Volumes and Surfaces of Solids of Revolution, Double Integral, Triple Integral and Dirichlet's Integral, Beta and Gamma Functions, and Differential Equations. Each unit is designed to build a comprehensive understanding of calculus and its applications.

• The book is divided into 16 units covering all major topics in calculus and differential equations.
• Each unit is focused on a specific area such as series, mean value theorems, curvature, partial differentiation, and more.
• The structure allows for systematic learning from basic to advanced concepts.
• 📌 Series: A sum of the terms of a sequence.
• 📌 Curvature: A measure of how sharply a curve bends.
• 📌 Partial Differentiation: Differentiation of functions with several variables with respect to one variable, keeping others constant.

Unit 1: Series - Outline of the Unit

This section provides the outline for Unit 1: Series. It lists all the subtopics that will be covered in this unit, including definitions, types of series, sequence of partial sums, convergence and divergence of series, important properties of infinite series, necessary conditions for convergence, convergence of geometric series, comparison tests, convergence of hyper-harmonic series, important 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-assessment questions, and exercise questions.

• Unit 1 will cover the definition and types of series.
• It will discuss convergence, divergence, and oscillatory behavior of series.
• The unit includes important tests for convergence such as Cauchy's root test, D'Alembert's ratio test, logarithmic ratio test, Raabe's test, De Morgan and Bertrand's test, Cauchy condensation test, and Gauss's test.
• Alternating series and their convergence will be discussed, including Leibniz's test.
• Absolute and conditional convergence will be explained.
• 📌 Series: A sum of the terms of a sequence.
• 📌 Convergence: When the sum of an infinite series approaches a finite value.
• 📌 Divergence: When the sum of an infinite series does not approach a finite value.