Advanced 𝑥 Algebra Topic 87 of 100
Infinite Geometric Series
Converges when |r|<1.
Key idea
Converges when |r|<1.
In depth
When |r| < 1, terms shrink to zero fast enough for the total to be finite: S = a/(1 − r). If |r| ≥ 1, the series diverges.
Worked example
1 + 1/2 + 1/4 + ... = 1/(1 − 1/2) = 2.
Formula
S = a/(1−r)
Real-world uses
- Explaining why repeating decimals equal fractions and how a bouncing ball’s total distance is finite
Common mistake
Applying the formula when |r| ≥ 1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Infinite Geometric Series?
Converges when |r|<1. It is a topic at the advanced level in Algebra.
What is the formula for Infinite Geometric Series?
The key formula is S = a/(1−r).
Where is Infinite Geometric Series used in real life?
Real-world examples: Explaining why repeating decimals equal fractions and how a bouncing ball’s total distance is finite.
What should I learn before Infinite Geometric Series?
Make sure you are comfortable with Geometric Series, the previous topic in Algebra.
What is a common mistake with Infinite Geometric Series?
Applying the formula when |r| ≥ 1.
How can I practise Infinite Geometric Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Infinite Geometric Series?
Move on to Sigma Notation next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.