Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
More Math Help
Tutoring
Tools and Resources
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books

You CAN Ace Calculus

17calculus > infinite series > geometric series

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

related topics

For a discussion of geometric sequences, see the page on sequences

A main application of a geometric series is to build power series

WikiBooks - Geometric Series

Geometric Series

Geometric Series are an important type of series that you will come across while studying infinite series. This series type is unusual because not only can you easily tell whether a geometric series converges or diverges but, if it converges, you can calculate exactly what it converges to. This is extremely unusual for an infinite series. Let's break down what this theorem is saying and how we can use it.

Geometric Series Convergence

A series in the form $$\displaystyle{ \sum_{n=0}^{\infty}{r^n}}$$ is called a Geometric Series with ratio $$r$$.

$$0 \lt \abs{r} \lt 1$$

$$\abs{r} \geq 1$$

converges

diverges

If the series converges, it converges to $$\displaystyle{\frac{1}{1-r}}$$.

This is usually written $$\displaystyle{ \sum_{n=0}^{\infty}{r^n} = \frac{1}{1-r}, ~~ 0 \lt \abs{r} \lt 1 }$$.

### Quick Summary

 used to prove convergence yes used to prove divergence yes can be inconclusive no can find convergence value yes useful for finding power series

How To Use The Geometric Series

A geometric series looks like this $$\displaystyle{ \sum_{n=0}^{\infty}{ r^n } }$$ where r is an expression of some sort, not containing n.

If you can get your series into this form using algebra, then $$r$$ will tell you whether the series converges or diverges.
If $$\abs{r} \geq 1$$, then the series diverges.
If $$\abs{r} < 1,$$ then the series converges and it converges to $$\displaystyle{\frac{1}{1-r}}$$

Notes

1. r is called the ratio.
2. The absolute values on r to determine convergence or divergence are absolutely critical. Do not drop them unless you are sure that r is positive all the time. If you are not sure, keeping them is always correct.
3. Watch out! - - Be careful to notice that the series given above starts with index $$n=0$$. When determining the convergence value, make sure you take that into account and adjust your series to exactly match this one, including the starting index value of zero.

Finite Geometric Series

In the following discussion, we are assuming that $$0 < r < 1$$.
The equation for the value of a finite geometric series is
$$\displaystyle{ \sum_{n=0}^{k}{r^n} = \frac{1-r^{k+1}}{1-r} ~~~~~ (1) }$$
where $$k$$ is a finite positive integer.

Practice 1

Show that the above formula holds for $$k=1, k=2$$ and $$k=3$$.

solution

Derivation

In this section, we will derive equation (1).
Let $$\displaystyle{ S_n = \sum_{k=0}^{n}{r^k} = }$$ $$1+r+r^2+r^3+ . . . + r^n$$
Multiplying this by $$r$$ gives us $$rS_n = r + r^2 + r^3 + . . . +r^n + r^{n+1}$$
Now we subtract $$rS_n$$ from $$S_n$$ to get $$S_n - rS_n = (1+ r + r^2 + r^3 + . . . +r^n)$$ $$-$$ $$(r+r^2+r^3+ . . . + r^n + r^{n+1} )$$
Looking closely at the terms on the right side of the equal sign, we can see that all the terms cancel except for $$1-r^{n+1}$$. So now we have $$S_n-rS_n = 1-r^{n+1}$$. Now we solve for $$S_n$$.
$$\begin{array}{rcl} S_n-rS_n & = & 1-r^{n+1} \\ S_n(1-r) & = & 1-r^{n+1} \\ S_n & = & \displaystyle{ \frac{1-r^{n+1}}{1-r} } \end{array}$$

$$\displaystyle{ \sum_{k=0}^{n}{r^k} = \frac{1-r^{k+1}}{1-r} }$$ [qed]

## Financial Application of Geometric Series

 This video shows a geometric series applied to a financial equation. The application of calculus to finances is not covered on this site. But you may find this video helpful if you are in business calculus.

### Search 17Calculus

Practice Problems

Instructions - - Unless otherwise instructed, for each of the following series
1. determine whether it converges or diverges, using the geometric series test, if possible.
2. If it converges, determine what it converges to (if possible). Give your answers in exact form.

 Level A - Basic

Practice A01

$$\displaystyle{\sum_{n=0}^{\infty}{\frac{3}{4^n}}}$$

solution

Practice A02

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2^n}{e^n} } }$$

solution

Practice A03

$$\displaystyle{\sum_{n=1}^{\infty}{\left[\frac{1+\sqrt{5}}{2}\right]^n}}$$

solution

Practice A04

Express $$\displaystyle{ 1+0.4+0.16+0.064+ . . . }$$ using sigma notation and determine convergence or divergence.

solution

Practice A05

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\pi^n }{ 3^{n+2}} } }$$

solution

Practice A06

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{ 3^n+2^n }{ 6^n } } }$$

solution

Practice A07

Express this sum using sigma notation
$$\displaystyle{ 1+0.1+0.01+0.001+0.0001+ . . . }$$

solution

Practice A08

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{ 1 }{ 5^n } } }$$

solution

Practice A09

Express this sum using sigma notation
$$\displaystyle{ 2/13 - 4/13^2 + 8/13^3 - 16/13^4 + . . . }$$

solution

Practice A10

$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{3^n}{4^n} + \frac{2}{7^n} \right] } }$$

solution

Practice A11

$$\displaystyle{ \sum_{k=0}^{\infty}{ \left( \frac{2}{3} \right)^k } }$$

solution

Practice A12

$$\displaystyle{ 1 + e^{-1} + }$$
$$\displaystyle{ e^{-2} + e^{-3} + . . . }$$

solution

Practice A13

$$\displaystyle{ \frac{3}{2} - \frac{3}{4} + \frac{3}{8} - \frac{3}{16} + . . . }$$

solution

Practice A14

$$\displaystyle{ 1 - \frac{1}{5} + \left( \frac{1}{5} \right)^2 - \left( \frac{1}{5} \right)^3 + . . . }$$

solution

 Level B - Intermediate

Practice B01

$$\displaystyle{\sum_{n=1}^{\infty}{\frac{(-1)^n2^{n-1}}{3^n}}}$$

solution

Practice B02

$$\displaystyle{ \sum_{n=3}^{\infty}{ 5 (2/3)^{n-1} } }$$

solution

Practice B03

$$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{ \pi^{n+1} }{ 6^n } } }$$

Express $$\displaystyle{ 0.\overline{21} }$$