# Book:Clement V. Durell/Advanced Algebra/Volume I

## Clement V. Durell: Advanced Algebra, Volume $\text { I }$

Published $\text {1932}$, Bell.

### Contents

Preface
$\text{I}$. Permutations and Combinations
(Arrangements of Unlike Things; Like and Unlike Things; Selections; Distribution in Groups; Selections, any number at a time)
$\text{II}$. Binomial Theorem: Positive Integral Index
(Expansion of $\paren {x + a}^n$; Greatest Coefficients and Terms; Relations between Coefficients; Summation of Series)
$\text{III}$. Finite Series
(Functional and $\Sigma$ Notation; Difference Method; Induction; Power Series)
Test papers. $\text{A}$. 1-10
$\text{IV}$. Limits and Convergence
(Idea of a Limit; Non-existence of Limits; Behaviour of $x^n$ when $n \to \infty$; Convergent, Divergent, Oscillating Series; General Properties of Limits; General Theorems on Convergence; Comparison Test; $\sigma n^{-p}$; Ratio Test; Positive and Negative Terms; Absolute Convergence)
$\text{V}$. The Binomial Series
( Conditions for Convergence; Greatest Terms; Notation for Coefficients; Binomial Theorem; Expansion of Functions; Forms of Binomial Series; Partial Fractions; Approximations; Homogeneous Products; Sum of First $r$ Coefficients)
$\text{VI}$. Logarithmic and Exponential Functions
(Properties of $\ds \int_1^t x^{-1} d x$; Natural Logarithms; $\map {\dfrac d {d x} } {\log x}$; Logarithmic Series; Common Logarithms; Proportional Parts; Summation of Series; Approximations; The Function $e^x$; Expansion of $e^x$; Value of $e$; $e$ is not Rational; Remainder after $n$ Terms of Exponential Series; $a^x$; $\ds \lim_{n \to \infty} \paren {1 + x/n}^n$)
Test papers. $\text{A}$. 11-20
$\text{VII}$. Rational Functions
(Quadratic Function; Cubic Function; Polynomials; Repeated Roots; $\paren {a x^2 + b x + c} / \paren {A x^2 + B x + C}$;
$\text{VIII}$. Theory of Equations
(Equations with Given Roots; Relations between Roots and Coefficients; Unsymmetrical Relations between Roots; Transformation of Equations; Numerical Equations; Newton's Method, Horner's Method; Elimination)
$\text{IX}$. Determinants
(Second Order Determinants; Third Order Determinants; Minors and Cofactors; Factorisation; Linear Equations; Product of Determinants; Determinants of any Order)
Test papers. $\text{A}$. 21-35