Sum of Infinite Geometric Series
Sum: 0
🔁 Sum of Infinite Geometric Series Calculator
✅ What Is an Infinite Geometric Series?
An infinite geometric series is a sequence of numbers that continues forever, where each term is multiplied by a constant common ratio (r) to get the next term.
📌 Example:
2 + 1 + 0.5 + 0.25 + ...
If the common ratio is between -1 and 1, the sum of this infinite series converges to a finite number.
📐 Formula to Calculate the Sum:
S=a1−rS = \frac{a}{1 - r}S=1−ra
Where:
- S = Sum of the infinite series
- a = First term of the series
- r = Common ratio (must be between -1 < r < 1)
🧮 Example:
First Term (a) = 5
Common Ratio (r) = 0.5 S=51−0.5=50.5=10S = \frac{5}{1 - 0.5} = \frac{5}{0.5} = 10S=1−0.55=0.55=10
✅ The infinite series converges to 10.
❗ When Does the Series Converge?
The formula is only valid when: −1<r<1-1 < r < 1−1<r<1
If |r| ≥ 1, the series diverges — i.e., it does not have a finite sum.
🧰 Features of the Calculator:
- Instant result using the formula
- Validates if the series converges
- Supports decimal and fractional input
- Works on mobile and desktop
- Easy for students, teachers, and researchers
👥 Who Can Use This?
- Students learning sequences and series
- Teachers creating math problems or worksheets
- Engineers and Physicists solving convergence problems
- Programmers handling iterative algorithms
- Anyone needing quick math verification
🔍 Frequently Asked Questions (FAQs)
❓ What is the common ratio in a geometric series?
It’s the fixed number you multiply each term by to get the next one.
❓ Can I use this formula if r = 1 or r > 1?
No. The series diverges and doesn't have a finite sum if |r| ≥ 1.
❓ What if the common ratio is negative?
You can use the formula as long as |r| < 1. It will alternate between positive and negative terms but still converge.
❓ Can the sum be negative?
Yes. If the first term or ratio is negative, the sum can be negative.
❓ How do I find the common ratio?
Divide the second term by the first term.
🎯 Final Thought
An infinite geometric series may go on forever, but its sum doesn’t have to. Use this calculator to instantly find the limit of your series and understand whether it converges or diverges.