Algebra & Trigonometry

Table of Content

Chapter 1  Infinite Series
1.1  Introduction
1.2  Definitions
1.3 General Properties of Series
1.3.1  Comparison Test (Form I)
1.3.2  Convergence of the Series ∑_(n=1)^∞▒1/n!
1.3.3  Convergence of the p− series  ∑_(n=1)^∞▒1/np
1.3.4  Cauchy’s Root Test
1.4  D’ Alambert’s Ratio Test
1.5  Alternating Series
1.5.1  Leibnitz Test for Convergence of an Alternating Series
1.5.2  Absolute and Conditional Convergence
Chapter 2  Binomial, Exponential and Logarithmic Series
2.1  Binomial Series
2.2  Some Standard Binomial Expansions
2.3  Expansions and Approximations
2.4  Summation of Series
2.5  Exponential Series
2.5.1 Exponential Theorem
2.6  Standard Exponential Expansions
2.7  Expansions and Approximations
2.8  Summation of Series
2.9  Logarithmic Series
2.9.1  Logarithmic Theorem
2.9.2  Some Standard Expansions
2.9.3  Expansion, Approximation and Summation
Chapter 3  Matrices
3.1  Introduction
3.2  Rank of a Matrix
3.3  Vectors
3.3.1  Addition of Vectors
3.3.2  Scalar Multiplication of a Vector
3.3.3  Linear Combination of Vectors
3.3.4  Linear Dependence and Linear Independence of Vectors
3.3.5  Methods of Testing Linear Dependence or
Independence of a Set of Vectors
3.4  Consistency of a System of Linear Algebraic Equations
3.4.1  Rouche’s Theorem
3.4.2  System of Homogeneous Linear
3.5  Eigenvalues and Eigenvectors
3.5.1  Properties of  Eigenvalues
3.6  Cayley − Hamilton Theorem
3.7  Similar Matrices
3.8  Diagonalisation of a Matrix
3.8.1  Calculation of Power of a Matrix A
3.8.2  Diagonalisation by Orthogonal Transformation or
Orthogonal Reduction
Chapter 4  Theory of Equations
4.1 Introduction
4.2  Relations between the Roots and the Coefficients of an
Algebraic Equation
4.3  Transformation of Equations
4.4  Reciprocal Equations
4.5  Numerical Methods for Approximate Roots
4.5.1  Horner’s Method
4.5.2  Newton’s Method
Chapter 5  Expansions of Trigonometric Functions
5.1  Expansions of cos nθ and sin nθ
5.2  Expansion of tan nθ in Powers of tan θ
5.3  Expansions of cos θ and sin θ in Ascending Powers of θ
5.4  Expansion of tan θ in Powers of θ
5.5  Expansions of cosn θ and sinn θ
Chapter 6  Hyperbolic Functions and Logarithms of
Complex Numbers
6.1  Exponential Forms of Circular Functions
6.2  Euler’s Formula
6.3  Hyperbolic Functions
6.4  Relations between Circular Functions and Hyperbolic Functions
6.5  Formulas Involving Hyperbolic Functions
6.6  Inverse Hyperbolic Functions
6.7  Logarithms of Complex Numbers

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