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Geometric Series Convergence and Divergence: A Complete Guide

Essential Maths: Master Core Skills for AU Students

Adding and Subtracting Fractions

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Using models to add and subtract fractions

Add Fractions With Models

Subtract Fractions With Models

Adding fractions with like denominators

Adding Fractions — with like denominators — 3-digit without simplifying

Adding Fractions — with like denominators — 3-digit

Adding Fractions — Word problem: baking

Add fractions with like denominator

Adding Fractions — with like denominators — 1-digit

Adding Fractions — with like denominators — 2-digit

Adding Fractions — Word problem: beads

Subtracting fractions with like denominators 

Subtract fractions with like denominator

Subtracting Fractions — with like denominators — 2-digit

Subtracting Fractions — with like denominators — Larger 2-digit

Subtracting Fractions — with like denominators — 3 digit

Subtracting fractions — Word problem: competition

Subtracting fractions — Word problem: snack

Adding and subtracting fractions with unlike denominators

Adding and Subtracting Fractions — Find the lowest common denominator (LCD)

Adding and Subtracting Fractions — Find the LCD and evaluate

Add fractions with unlike denominator

Subtract fractions with unlike denominator

Adding and Subtracting Fractions — Word problem: cargo ship

Adding Fractions — with unlike denominators

Subtracting Fractions — with unlike denominators

Adding and subtracting mixed numbers

Adding Mixed Numbers — Word Problem

Subtracting Mixed Numbers — Word Problem

Add mixed fractions with like denominator

Add mixed fractions with unlike denominator

Multiplying and Dividing Fractions

Multiplying fractions and whole numbers

Multiply a fraction with a whole number

Multiplying Fractions — with whole numbers — 1 digit

Multiplying Fractions — with whole numbers — 2 digits

Application of Multiplying Fractions and Whole Numbers – Word Problems

Dividing fractions with whole numbers

Divide a fraction by a whole number

Divide a Fraction by a Whole Number

Divide a Fraction by a Whole Number Number RHS

Multiplying proper fractions

Multiply Proper Fractions — 1 Digit Fractions

Multiplying improper fractions and mixed numbers

Multiply Mixed Numbers

Dividing fractions and mixed numbers

Divide Fractions and Mixed Numbers – Word Problem

Dividing Mixed numbers

Dividing Mixed Numbers

Applications of fraction operations

Fraction Operations

Pythagorean Theorem

Squares and square roots

Square Root of a Perfect Square

Pythagorean theorem

Estimating square roots

Using the pythagorean relationship

Applications of pythagorean theorem

Percents

Taxes, discounts, tips and more

Discounts — Word Problem

Taxes — Word Problem

Percentage - Applying a Discount Price– Word Problem

Percentage - Applying a Discount Percent – Word Problem

Percentage - Applying a Tax Total – Word Problem

Percentage - Applying a tax Percent – Word Problem

Simple interest

Simple Interest — Word Problem

Basic Concepts of Simple Interest – Word Problem

Borrowing Money for More Than One Year  – Word Problem

Investigate the Advantages of Borrowing Money  – Word Problem

Representing percents

Representing Percents

Representing Percents — Word Problem

Representing Percents Two Graphs – Word Problem

Percents, fractions, and decimals

Conversion — Decimal to Fraction

Conversion — Decimal to Percent

Conversion — Fraction to Decimal

Conversion — Fraction to Percent

Conversion — Percent to Decimal

Conversion — Percent to Fraction

Percents – Word Problem

Convert a Decimal to a Fraction

Convert a Decimal to a Percent

Convert a Fraction to a Decimal

Convert a Fraction to a Percent

Convert a Percent  to a Decimal

Convert a Percent to a Fraction

Percents Fractions And Decimals – Word Problem

Percent of a number

Percent of a Number

percent of a number – Word Problem

Percent of a Number  Fraction Percent

Percent of a Number Integer Percent 

Percent of a number number With Decimal Percent 

Percent of a Number  Three Digits Number Percent

Adding and multiplying percents

Add and Multiply Percents — Word Problem

Add and Multiply Percents – Word Problem

Surds

Evaluating and simplifying surds

Evaluating and simplifying radicals

Square and square roots

Cubic and cube roots

Cube root of a perfect cube

Converting radicals to mixed radicals

Converting radicals to entire radicals

Adding and subtracting surds

Adding and subtracting radicals

Adding radicals

Subtracting radicals

Multiplying and dividing surds

Multiplying and dividing radicals

Dividing radicals

Multiplying radicals

Rationalize the denominator 

Algebraic Expressions

Equivalent algebraic expressions

Equivalent expressions of polynomials

Adding and subtracting algebraic expressions

Adding and subtracting polynomials

Multiplying monomial by monomial

Multiply a Monomial by a Monomial

Multiplying monomial by binomial

Multiplying binomial by binomial

Multiplying polynomial by polynomial

Applications of polynomials

Introduction to Relations and Functions

Relationship between two variables

Understand relations between x- and y-intercepts

Domain and range of a function

Identifying functions

Function notation

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Gradient equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Gradient intercept form: y = mx + b

Slope intercept form: y = mx + b

General form: Ax + By + C = 0

Gradient-point form: \(y - y_1 = m (x - x_1)\)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and gradient

Word problems of graphing linear functions

Perpendicular line equation

Rate of change

Parallel and perpendicular lines in linear functions

Applications of linear relations

Simultaneous Equations

Solving 3 variable simultaneous equations by substitution

Solving 3 variable systems of equations by substitution

Solving 3 variable simultaneous equations by elimination

Solving 3 variable systems of equations by elimination

Word problems relating 3 variable simultaneous equations

Aligned with your class and textbook, our comprehensive help on Essential Mathematics covers all topics that you would see in your maths class, whether it&apos;s the Australian curriculum for <a href="http://www.australiancurriculum.edu.au/seniorsecondary/mathematics/essential-mathematics/curriculum/seniorsecondary#page=1">Essential Mathematics</a>, <a href="http://wace1516.scsa.wa.edu.au/assessment/externally-set-tasks">Externally Set Tasks</a> (EST), <a href="http://wace1516.scsa.wa.edu.au/assessment/examinations/examination-information">ATAR</a> course exam.

In the video lessons, our Essential maths tutors walk you through all topics like Trigonometry, Simultaneous equations, Operations with Surds, Derivatives, Congruent triangles, Linear programming, and more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest essential maths problems. Then, reinforce your understanding with tons of essential maths practice.

All our lessons are taught by experienced essential maths teachers. Let&apos;s finish your homework in no time, and ACE that final.

Essential Maths

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https://dmn92m25mtw4z.cloudfront.net/img/textbook/au/textbook-au-essential-maths.png

Mastering Geometric Series Convergence and Divergence

What You'll Learn

Identify when a geometric series converges or diverges based on the common ratio r

Apply the formula a/(1-r) to find the sum of convergent geometric series

Recognize geometric series in various algebraic forms and manipulate them to standard form

Determine the value of r and verify it falls between -1 and 1 for convergence

Show that geometric series with |r| 1 diverge to infinity

What You'll Practice

Algebraically manipulating series to reveal geometric form with factoring and exponent laws

Finding sums of convergent geometric series starting at n=0 or n=1

Converting series indices to match standard geometric series formulas

Proving divergence when the common ratio exceeds 1 in absolute value

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

8 Video lessons

Practice exercises

Learning resources

Skills

Geometric Series

Convergence

Divergence

Infinite Series

Common Ratio

Series Summation

Algebraic Manipulation

Exponent Laws