Formula Used
A geometric series has first term a and common ratio r. Each term is multiplied by r.
S∞ = a / (1 - r), when |r| < 1.
Sn = a(1 - r^n) / (1 - r), when r ≠ 1.
Remainder = S∞ - Sn. The absolute remainder estimates the remaining tail.
If |r| ≥ 1, the infinite geometric sum is divergent. The calculator still reports the finite partial sum.
How to Use This Calculator
Enter the first term of the series. Then enter the common ratio. Use a negative ratio for alternating terms.
Add the number of terms for the partial sum. Choose a tolerance for the error target. Set decimal precision for the displayed answer.
Press Calculate to view the convergence test, infinite sum, partial sum, remainder, and needed terms. Use the export buttons to save the result.
Example Data Table
| First term |
Ratio |
Terms |
Infinite sum |
Status |
| 1 |
0.5 |
10 |
2 |
Convergent |
| 3 |
0.25 |
8 |
4 |
Convergent |
| 5 |
-0.5 |
12 |
3.333333 |
Convergent |
| 2 |
1.1 |
10 |
Not applicable |
Divergent |
Understanding Infinite Series
An infinite series adds terms without a final stopping point. Many useful series still have a fixed total. This happens when later terms become small fast enough. The calculator focuses on geometric series, because they have a clear convergence rule. A geometric series starts with a first term. Each next term is made by multiplying the previous term by a common ratio. If the absolute ratio is less than one, the terms shrink toward zero. The sum then approaches one finite limit.
Why Convergence Matters
Convergence tells you whether an infinite total is meaningful. A ratio of one keeps repeating the same size. A ratio greater than one grows instead of fading. A ratio of negative one or below can oscillate or expand. In these cases, the infinite sum does not settle. The tool marks them as divergent, so the result is not mistaken for a real total.
Partial Sums and Accuracy
A partial sum adds only a chosen number of terms. It is useful when you need an approximation. The calculator compares that partial sum with the infinite sum when convergence exists. The difference is the remainder estimate. Smaller ratios usually reduce the remainder quickly. More terms also improve accuracy. A tolerance field helps estimate how many terms are needed before the remaining error is small enough.
Practical Uses
Infinite series appear in finance, physics, computer science, and statistics. They help model repeated discounts, signal decay, probability tails, and recursive processes. They also support classroom work where learners test formulas against visible examples. Export options make the result easier to keep, share, or place in a report.
Good Input Habits
Use decimal ratios when needed. Enter negative ratios for alternating series. Choose a realistic term count for partial sums. Use a small positive tolerance when you need an accuracy target. Review the convergence message before using the final sum. A divergent series may still have finite partial sums, but it has no infinite total.
Example Checks
For example, one half plus one quarter plus one eighth keeps shrinking. Its limit is one. The calculator shows both the exact infinite result and a chosen partial result. This makes the idea easier to verify step by step clearly.
FAQs
1. What series does this calculator handle?
It handles infinite geometric series. You enter a first term and common ratio. The tool checks convergence, finds the infinite sum when possible, and reports useful partial sum details.
2. When does an infinite geometric series converge?
It converges when the absolute value of the common ratio is less than one. That means each later term becomes smaller and the total approaches a finite limit.
3. What happens if the ratio is one?
The terms do not shrink. The infinite total does not settle to a finite value. The calculator reports the finite partial sum only and marks the infinite series as divergent.
4. Can I use a negative ratio?
Yes. A negative ratio creates an alternating geometric series. It still converges if the absolute value of the ratio is less than one.
5. What is a partial sum?
A partial sum adds a fixed number of terms. It is an approximation of the infinite sum when the series converges. More terms usually improve the approximation.
6. What does the remainder mean?
The remainder is the difference between the infinite sum and the selected partial sum. Its absolute value shows the estimated remaining error after the chosen number of terms.
7. What is the tolerance field for?
Tolerance sets your desired error limit. The calculator estimates how many terms are needed before the remaining geometric tail is within that limit.
8. Can I download the answer?
Yes. After entering values, use the CSV or PDF buttons. The saved file includes the main inputs, convergence status, sums, remainder, and term estimate.