Calculator
Example Data Table
This sample uses n = 6 for the partial sum column.
| First term (a₁) | Ratio (r) | Infinite sum | Partial sum S₆ |
|---|---|---|---|
| 2 | 0.5 | 4 | 3.9375 |
| 5 | -0.2 | 4.1666666667 | 4.1664 |
| -3 | 1/3 | -4.5 | -4.4938271605 |
| 10 | 0.8 | 50 | 36.8928 |
Formula Used
Infinite sum: S∞ = a₁ / (1 - r)
Convergence condition: |r| < 1
Partial sum: Sₙ = a₁(1 - rⁿ) / (1 - r), when r ≠ 1
n-th term: aₙ = a₁rⁿ⁻¹
If |r| is not less than 1, the infinite series does not settle to a fixed total.
How to Use This Calculator
- Enter the first term of the geometric series.
- Choose how you want to provide the common ratio.
- Type the ratio as a decimal, percentage, fraction, or derive it from the first two terms.
- Optionally enter n to calculate a partial sum and the n-th term.
- Choose how many early terms you want to preview.
- Press calculate to show the result above the form.
- Use the CSV or PDF buttons to save the result set.
About Infinite Geometric Series
What this calculator does
An infinite geometric series adds terms that keep multiplying by the same common ratio. The first term sets the starting value. The ratio controls how fast the terms shrink, grow, or alternate. This calculator helps you test convergence first. It then computes the infinite sum when that sum exists. It also returns a partial sum, the n-th term, a remainder estimate, and a preview of early terms. These outputs support algebra lessons, exam revision, and sequence analysis. They also help teachers explain why some infinite processes still produce a finite answer.
Why convergence matters
The key rule is simple. An infinite geometric series converges only when the absolute value of the common ratio is less than one. In that case, later terms move toward zero. The running total gets closer to a fixed limit. If the ratio is 1, the terms never shrink. If the ratio is greater than 1 in magnitude, the terms grow or keep swinging without settling. This calculator highlights that condition clearly. It shows whether the infinite sum is valid before reporting a final answer.
Flexible ratio input for better practice
Students do not always receive the ratio in decimal form. Sometimes it appears as a percentage. Sometimes it is given as a fraction. In other cases, you only know the first two terms. This page handles all of those input styles. That makes it useful for homework, quizzes, and worksheet checking. It also reduces conversion mistakes. You can compare the same series in different formats and confirm that every method leads to the same ratio and the same convergent sum.
Use cases in maths and beyond
Infinite geometric series appear in algebra, finance, signal models, repeating decimals, and growth or decay analysis. Alternating ratios are also common. The calculator helps you inspect those patterns quickly. You can see how the early terms behave. You can compare a partial sum with the full limiting sum. That is useful when learning approximation error. With CSV and PDF export, results are easier to save for class notes, worked examples, or quick review before tests.
FAQs
1) What is an infinite geometric series?
It is a series where each term is found by multiplying the previous term by the same common ratio. The series continues forever, but it can still have a finite sum when the ratio meets the convergence rule.
2) When does the infinite sum exist?
The infinite sum exists only when the absolute value of the common ratio is less than one. That condition makes the terms shrink toward zero, so the running total approaches a fixed limit.
3) What formula is used for the infinite sum?
The calculator uses S∞ = a₁ / (1 - r). Here a₁ is the first term and r is the common ratio. This formula works only for convergent geometric series.
4) Can the ratio be negative?
Yes. A negative ratio creates alternating positive and negative terms. The series still converges if the absolute value of the ratio is less than one.
5) What happens if |r| is 1 or more?
Then the infinite series does not converge to one fixed number. The terms may stay constant, grow larger, or keep oscillating. In that case, the infinite sum is not reported.
6) Why would I calculate a partial sum too?
A partial sum shows the total after a finite number of terms. It helps you compare approximation against the full convergent sum and understand how quickly the series approaches its limit.
7) Can I find the ratio from the first two terms?
Yes. If you know a₁ and a₂, then r = a₂ / a₁, provided the first term is not zero. This option is useful in reverse problems and pattern recognition exercises.
8) Where are infinite geometric series used?
They appear in algebra, repeating decimals, compound models, physics, finance, and signal analysis. They are also useful for teaching convergence, approximation, and the behavior of recursive patterns.