How to Solve Infinite Geometric Series

How to Solve Infinite Geometric Series

Learn how to solve the Infinite Geometric Series using the following step-by-step guide and examples.

Related Topics

Step by step guide to solve Infinite Geometric Series

  • Infinite Geometric Series: The sum of a geometric series is infinite when the absolute value of the ratio is more than \(1\).
  • Infinite Geometric Series formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}}\)

Infinite Geometric Series – Example 1:

Evaluate infinite geometric series described. \(S= \sum_{i=1}^ \infty 9^{i-1}\)

Solution:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{i=1}^ \infty 9^{i-1}=\frac{1}{1-9}=\frac{1}{-8}=-\frac{1}{8}\)

Infinite Geometric Series – Example 2:

Evaluate infinite geometric series described. \(S= \sum_{k=1}^ \infty (\frac{1}{4})^{k-1}\)

Solution:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{k=1}^ \infty (\frac{1}{4})^{k-1}=\frac{1}{1-\frac{1}{4}}=\frac{1}{\frac{3}{4}}=\frac{4}{3}\)

Infinite Geometric Series – Example 3:

Evaluate infinite geometric series described. \(S= \sum_{i=1}^ \infty 8^{i-1}\)

Solution:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{i=1}^ \infty 8^{i-1}=\frac{1}{1-8}=\frac{1}{-7}=-\frac{1}{7}\)

Infinite Geometric Series – Example 4:

Evaluate infinite geometric series described. \(S= \sum_{k=1}^ \infty (\frac{1}{2})^{k-1}\)

Solution:

Use this formula: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{k=1}^ \infty (\frac{1}{2})^{k-1}=\frac{1}{1-\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2\)

Exercises for Solving Infinite Geometric Series

Determine if each geometric series converges or diverges.

  • \(\color{blue}{a_{1} = –3, r = 4}\)
  • \(\color{blue}{a_{1}= 5.5, r = 0.5}\)
  • \(\color{blue}{a_{1} = –1, r = 3}\)
  • \(\color{blue}{81 + 27 + 9 + 3 …,}\)
  • \(\color{blue}{–3 + \frac{12}{5} – \frac{48}{25} + \frac{192}{125} …,}\)
  • \(\color{blue}{\frac{128}{3125} – \frac{64}{625} + \frac{32}{125} – \frac{16}{25 }…,}\)

Answers

  • \(\color{blue}{Diverges}\)
  • \(\color{blue}{Converges}\)
  • \(\color{blue}{Diverges}\)
  • \(\color{blue}{Converges}\)
  • \(\color{blue}{Converges}\)
  • \(\color{blue}{Diverges}\)

Related to "How to Solve Infinite Geometric Series"

Best Tablet Floor Stands For Online Teaching
Best Tablet Floor Stands For Online Teaching
ASVAB Arithmetic and Mathematics Preview
ASVAB Arithmetic and Mathematics Preview
What Skills Do I Need for the ACCUPLACER Math Test?
What Skills Do I Need for the ACCUPLACER Math Test?
Online Math Tutoring Tools: The Top 5 tools
Online Math Tutoring Tools: The Top 5 tools
7 Best Headphones for Online Lessons
7 Best Headphones for Online Lessons
Top 20 Math Websites for Virtual Learning
Top 20 Math Websites for Virtual Learning
Math Skills You Need for the GED Math Test
Math Skills You Need for the GED Math Test
Top Ten Cameras for Classroom Recording
Top Ten Cameras for Classroom Recording
Top 6 Travel-Friendly Teaching Supplies for your Portable Classroom
Top 6 Travel-Friendly Teaching Supplies for your Portable Classroom
List Of the Best Middle School Math Supply for Learning
List Of the Best Middle School Math Supply for Learning

Leave a Reply

36% OFF

Download Instantly

X

How Does It Work?

Find Books

1. Find eBooks

Locate the eBook you wish to purchase by searching for the test or title.

add to cart

2. Add to Cart

Add the eBook to your cart.

checkout

3. Checkout

Complete the quick and easy checkout process.

download

4. Download

Immediately receive the download link and get the eBook in PDF format.

Why Buy eBook From Effortlessmath?

Save money

Save up to 70% compared to print

Instantly download

Instantly download and access your eBook

help environment

Help save the environment

Access

Lifetime access to your eBook

Test titles

Over 2,000 Test Prep titles available

Customers

Over 80,000 happy customers

Star

Over 10,000 reviews with an average rating of 4.5 out of 5

Support

24/7 support

Anywhere

Anytime, Anywhere Access

Find Your Test

Schools, tutoring centers, instructors, and parents can purchase Effortless Math eBooks individually or in bulk with a credit card or PayPal. Find out more…