About this topic
Before you begin
Use the H3 additional results: classical inequalities, congruences, the method of differences, reduction formulae, improper integrals, limits, bijections and inclusion–exclusion.
Learning goals
- Use and prove the AM–GM, Cauchy–Schwarz and triangle inequalities
- Reason with congruences and modular arithmetic
- Sum a series by the method of differences
- Derive and use reduction formulae for integrals
- Decide whether an improper integral converges and evaluate it
- Compare growth rates and combine limits
- Count with bijections and the inclusion–exclusion principle
Syllabus statements covered
- Include: Functions and Graphs concepts from H2 Mathematics; Sequences and Series concepts from H2 Mathematics, with the following addition: Summation of series by the method of differences; Complex Numbers concepts from H2 Mathematics; Calculus concepts from H2 Mathematics, with the following addition: Reduction formulae, Improper integrals; Probability (including counting) concepts from H2 Mathematics, with the following addition: Bijection principle (include the case of distributing indistinguishable objects into distinguishable boxes), Inclusion-Exclusion principle; Additional inequalities: AM-GM inequality, Cauchy-Schwarz inequality, Triangle inequality; Introduction to limits: Comparing polynomial, exponential, and logarithmic growth rates, Operations involving limits e.g., limit of a sum is the sum of the limits, if exist; Concepts of congruence and modular arithmetic.
Choose a lesson
Work through them in order.
- Not startedAM–GM, Cauchy–Schwarz and the triangle inequalityNextProve and apply the AM–GM, Cauchy–Schwarz and triangle inequalities, including their equality cases.
- Not startedCongruences and modular arithmeticNextCalculate with congruences and use residue classes to prove divisibility and the absence of integer solutions.
- Not startedDifferences, reduction formulae and limitsNextSum series by the method of differences, derive reduction formulae, evaluate improper integrals and compare growth rates.
- Not startedBijections and inclusion–exclusionNextCount by matching one set with another, distribute identical objects into boxes and correct overcounting with inclusion–exclusion.