Geometric Series Convergence
Converges iff |r| < 1.
Key idea
Converges iff |r| < 1.
In depth
Σ arⁿ converges to a/(1 − r) if |r| < 1, and diverges otherwise. It is the model against which many other series are compared.
Worked example
Σ from n=0 of (2/3)ⁿ = 1/(1 − 2/3) = 3.
Formula
Real-world uses
- Checking when a repeated discount or loop gain stays finite in economics and electronics
Common mistake
Applying the formula to a series that starts at n = 1 without adjusting.
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 Geometric Series Convergence?
Converges iff |r| < 1. It is a topic at the advanced level in Calculus.
What is the formula for Geometric Series Convergence?
The key formula is Σ rⁿ = 1/(1−r).
Where is Geometric Series Convergence used in real life?
Real-world examples: Checking when a repeated discount or loop gain stays finite in economics and electronics.
What should I learn before Geometric Series Convergence?
Make sure you are comfortable with Infinite Series, the previous topic in Calculus.
What is a common mistake with Geometric Series Convergence?
Applying the formula to a series that starts at n = 1 without adjusting.
How can I practise Geometric Series Convergence?
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 Geometric Series Convergence?
Move on to Divergence Test next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.