Sum to Infinity of a Geometric Series

We can often find the sum of the terms of an infinite geometric sequence rather than the sum of only the first $n$ terms.

Infinite series

An infinite series is the sum of the terms of an infinite sequence.
For example, $2+4+8+16+...$ is an infinite series, as is indicated by the " $+...$ ".
In sigma notation, $\sum_{k=1}^{\infty}2^k$ is an infinite series as the upper limit of $k$ is infinity.

Divergent

If the terms of a geometric series increase, then as the number of terms in the series increases to infinity, the value of the sum will also increase to infinity.
- We say that this series is divergent.
The value of the sum diverges away from being a fixed, known value.
All non-zero arithmetic series are divergent.

Convergent

If the terms of a geometric series decrease, then as the number of terms in the series increases to infinity, the value of the sum gets closer and closer to a fixed value.
- We say that this series is convergent.
The value of the infinite sum has a fixed value that we can find.
The more finite terms we have in the series, the closer the sum will be to that fixed value.

Common ratio

We can determine if a sequence is convergent by looking at the common ratio of the sequence, $r$ :
- A sequence is convergent if $|r|<1$ .
We can also write this as $-1<r<1$ .

Sum to infinity

The sum of the first $n$ terms of a geometric series is $S_n = \frac{\small a(1-r^n)}{\small 1-r}$ .
If $|r|<1$ , then as $n$ gets larger, the term $r^n$ gets smaller.
- This means for $n=\infty$ , $r^n=0$ .
The sum to infinity of a geometric series is equal to:
- $S_{\infty} = \frac{a}{1-r}$

Example

Determine whether the infinite geometric series with $a=27$ and $r=\frac{1}{3}$ is a divergent or convergent series.
- Find the sum of the infinite series if it is convergent.

Common ratio

If the infinite series is convergent, then $|r|<1$ .
Substituting in $r=\frac{1}{3}$ , we can see that:
- $\left|\frac{1}{3}\right|<1$
So the infinite series is convergent and we can find the sum.

Find the sum

The sum of a convergent infinite series is given by:
- $S_{\infty}=\frac{\small a}{\small 1-r}$
Substituting in $a = 27$ and $r=\frac{1}{3}$ , we get:
- $S_{\infty}=\frac{27}{\small 1-\frac{1}{3}}=\frac{27}{\small \frac{2}{3}} = 40.5$

1Proof

1.1Types of Numbers

1.1.1Whole Numbers

1.1.2The Four Rules for Positive Numbers

1.1.3Negative Numbers

1.1.4Integers

1.1.5The Four Rules for Negative Numbers

1.1.6Real Numbers

1.1.7Rational vs Irrational

1.2Notation

1.2.1Sets

1.2.2Subsets

1.2.3Venn Diagrams

1.2.4Sets Numbers

1.2.5Notation in Mathematical Arguments

1.3Proof

1.3.1Mathematical Proof

1.3.2Proof by Deduction

1.3.3Proving an Identity

1.3.4Polynomial Identities

1.3.5Proof by Exhaustion

1.3.6Disproof by Counter Example

1.3.7Proof by Contradiction (A2 Only)

2Algebra & Functions

2.1Powers & Roots

2.1.1Exponential Rules

2.1.2Rational Exponents

2.1.3Surds

2.1.4Simplify Surds

2.1.5Rationalise Denominator

2.2Quadratic Equations

2.2.1Quadratic +/-

2.2.2Factorise Quadratics

2.2.3Quadratic Formula

2.2.4Complete the Square

2.2.5Solve Quadratics Graphically

2.2.6Simultaneous Equations

2.2.7Simultaneous Quadratic Equations

2.2.8Simultaneous Equations Graphs

2.2.9Simultaneous Equations Word Problems

2.2.10Quadratic Equations of Powers

2.3Inequalities

2.3.1Solving Inequalities

2.3.2Solving Double Inequalities

2.3.3Solving Quadratic Inequalities

2.3.4Representing Inequalities

2.3.5Graphing Inequalities

2.3.6Set Notation

2.3.7Set Notation & Inequalities

2.4Polynomials

2.4.1Polynomials

2.4.2Adding & Subtracting Polynomials

2.4.3Remainder Theorem

2.4.4Algebraic Long Division

2.4.5Rewriting Rational Expressions

2.5Graphs

2.5.1Graphs of Polynomials

2.5.2Simultaneous Equations Graphs

2.5.3Simultaneous Quadratic Equations

2.5.4Graphs of Proportion

2.5.5Graphs of Common Functions 1

2.5.6Graphs of Common Functions 2

2.5.7Graphs of Common Functions 3

2.5.8Proportion

2.5.9Direct Proportion

2.5.10Inverse Proportion

2.6Functions

2.6.1Properties of Functions

2.6.2Inverse Functions

2.6.3Composite Functions

2.7Transformation of Graphs

2.7.1Reflections

2.7.2Rotation

2.7.3Translation

2.7.4Stretches & Compressions

2.7.5Effects of Transformations

2.7.6Combinations of Transformations

2.8Partial Fractions (A2 Only)

2.8.1Partial Fractions

2.8.2Partial Fractions with Three Linear Factors

2.8.3Partial Fractions with Three Linear Factors 2

2.8.4Partial Fractions with Repeated Factors

2.8.5Partial Fractions - Practice

3Coordinate Geometry

3.1Straight Lines

3.1.1y = mx +c

3.1.2Gradient & Y-intercept

3.1.3Distance-Time Graph

3.1.4Velocity-Time Graph

3.1.5Conversion Graph

3.2Circles

3.2.1Equations of Circles centred at Origin

3.2.2Finding the Centre & Radius

3.2.3Equation of a Tangent

3.2.4Circle Theorems - Perpendicular Bisector

3.2.5Circle Theorems - Angle at the Centre

3.2.6Circle Theorems - Angle at a Semi-Circle

3.2.7Equation of a Perpendicular Bisector

3.2.8Equation of a Circumcircle

3.2.9Circumcircle of a Right-angled Triangle

3.3Parametric Equations (A2 only)

3.3.1Parametric Equations

3.3.2Modelling with Parametric Equations

4Sequences & Series

4.1Binomial Expansion

4.1.1Pascal's Triangle

4.1.2Factorials

4.1.3Binomial Expansion

4.1.4Binomial Expansion & Rational Powers (A2 only)

4.1.5Binomial Expansion & Rational Powers 2 (A2 only)

4.2Sequences (A2 only)

4.2.1Nth term Rules

4.2.2Recurring Rules

4.2.3Types of Sequences

4.2.4Arithmetic Sequences

4.2.5Geometric Sequences

4.2.6Sigma Notation

4.2.7Arithmetic Series

4.2.8Geometric Series

4.2.9Sum to Infinity of a Geometric Series

4.2.10Modelling with Sequences & Series

5Trigonometry

5.1Using Angles

5.1.1Radians

5.1.2Arc length

5.1.3Area of Sectors

5.1.4Area of Segments

5.2Trigonometric Functions

5.2.1Unit Circle

5.2.2Properties of Trig Functions

5.2.3Trigonometry

5.2.4Exact values

5.2.5Graphs of Trigonometric Functions

5.2.6Small Angle Approximation

5.3Triangle Rules

5.3.1Sine Rules

5.3.2Cosine Rules

5.3.3Area of a Triangle

5.4Trigonometric Identities

5.4.1Secant, Cosecant & Cotangent

5.4.2Pythagorean Identities

5.4.3Angle Addition Formulae

5.4.4Manipulating the Angle Addition Formulae

5.4.5Double Angle Formula

6Exponentials & Logarithms

6.1Exponentials & Logarithms

6.1.1Exponential Function

6.1.2Logarithms

6.1.3Properties of Logarithms

6.1.4Natural Logarithm

6.1.5Graphs of Exponents & Logarithms

6.2Exponential Growth Decay

6.2.1Population Growth

6.2.2Depreciation

6.2.3Graphing Linear and Exponential Growth

6.2.4Interpreting Exponential Growth and Decay Graphs

6.2.5Compound Growth & Decay

6.2.6Compund Interest

7Differentiation

7.1Derivatives

7.1.1Gradient of a Curve

7.1.2Differentiation from First Principles

7.1.3Differentiation from First Principles - Practice

7.1.4Differentiation

7.1.5Finding derivatives

7.1.6Second Order Derivatives

7.2Graphs & Differentiation

7.2.1Stationary Points

7.2.2Maxima & Minima

7.2.3Modelling with Differentiation

7.2.4Using Second Derivatives (A2 only)

7.2.5Rate of Change (A2 only)

7.3Differentiation With Trigonometry and Exponentials

7.3.1Differentiation with Sin & Cos

7.3.2Diff. of Exponential Functions

7.3.3Differentiation of Trig. Functions 1 (A2 only)

7.3.4Differentiation of Trig. Functions 2 (A2 only)

7.4Rules of Differetiation (A2 only)

7.4.1Chain Rule

7.4.2Product Rule

7.4.3Quotient Rule

7.5Parametric & Implicit Differentiation

7.5.1Parametric Differentiation

7.5.2Implicit Differentiation

8Integration

8.1Integration

8.1.1Integration - Basics

8.1.2Integration - Definite Integrals

8.1.3Finding Functions

8.1.4Area under curves

8.1.5Area under x-axis

8.1.6Area between curves and lines

8.2Integration (A2 Only)

8.2.1Integration - Standard Functions

8.2.2Integration - Reverse Chain Rule

8.2.3Integration by Substitution

8.2.4Integration by Parts

8.2.5Integration by Parts - Practice

8.2.6Integration by Partial Fraction

8.2.7Modelling with Differential Equations

9Numerical Methods

9.1Finding Solutions

9.1.1Number Iterations

9.1.2Iterations to 1 Decimal Places

9.1.3Iteration Formula

9.1.4Locating Roots

9.1.5Newton Raphson Method

9.2Finding the Area

9.2.1Trapezium Rule

10Vectors

10.12D Vectors

10.1.1Vectors - Magnitude & Direction

10.1.2Unit Vectors

10.1.3Vectors - Algebraic Operations

10.1.4Position Vectors

10.1.5Vector Proofs

10.1.6Vectors Word Questions

10.23D Vectors

10.2.13D Vector Magnitude

10.2.23D Vectors - Algebraic Operations

10.2.3Position Vectors 3D

10.2.43D Vectors Word Questions

10.3Vector Proofs

10.3.1Coming Soon

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