How to Solve Infinite Geometric Series? (+FREE Worksheet!)
Learn how to solve the Infinite Geometric Series using the following step-by-step guide and examples.
A friendly Algebra 2 tutor note
Solve Infinite Geometric Series: how to make it click
Algebra 2 often looks harder because there are more symbols on the page. The good move is to slow down, identify the structure, and work one clean step at a time.
Good news: most Algebra 2 mistakes are small setup mistakes. If you can name the type of problem, you are already halfway to choosing the right tool.
Start here
Decide whether you are looking at a sequence of terms or a sum of terms. Then look for the common difference or common ratio.
Watch for this
Do not use a series formula before identifying whether it is arithmetic or geometric.
Two more tutor examples
Arithmetic sum
Example: 3 + 6 + 9 + 12
- There are 4 terms.
- Add directly or use the arithmetic-series idea.
- 3 + 6 + 9 + 12 = 30.
Answer: 30
Geometric pattern
Example: 2, 6, 18, 54
- Each term multiplies by 3.
- The common ratio is 3.
- Use that ratio for the next term.
Answer: 162
Try this quick confidence check
Try: Find the sum of 2 + 4 + 6 + 8.
Answer: 20.
When this feels steady, go back to the Algebra 2 hub and try the matching quiz or worksheet while the idea is still fresh.
Solve Infinite Geometric Series: pop-up practice
Try three quick questions. The goal is not perfection; it is noticing what you understand and what needs one more look.
Choose an answer to begin.
1. 5, 8, 11, 14 is:
2. 2, 10, 50 has common ratio:
3. Sigma notation means:
Related Topics
- How to Solve Finite Geometric Series
- How to Solve Geometric Sequences
- How to Solve Arithmetic Sequences
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: \(\color{blue}{S= \sum_{i=0}^ \infty a_{i}r^i=\frac{a_{1}}{1-r}} → S= \sum_{i=1}^ \infty 9^{i-1}=\frac{1}{1-9}=\frac{1}{-8}=-\frac{1}{8}\)
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Infinite Geometric Series – Example 2:
Evaluate the 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 the 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 the 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
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- \(\color{blue}{Converges}\)
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