Calculator Inputs
Example Data Table
This example uses a = 5, r = 0.4, and six preview terms.
| Term Number | Term Expression | Term Value | Partial Sum |
|---|---|---|---|
| 1 | 5 × 0.4^0 | 5.0000 | 5.0000 |
| 2 | 5 × 0.4^1 | 2.0000 | 7.0000 |
| 3 | 5 × 0.4^2 | 0.8000 | 7.8000 |
| 4 | 5 × 0.4^3 | 0.3200 | 8.1200 |
| 5 | 5 × 0.4^4 | 0.1280 | 8.2480 |
| 6 | 5 × 0.4^5 | 0.0512 | 8.2992 |
Because |0.4| is less than 1, the infinite series converges.
Formula Used
This calculator tests the ratio magnitude first. It then computes a preview partial sum. If the convergence condition holds, it also returns the infinite sum and a remainder estimate.
How to Use This Calculator
- Enter the first term, a.
- Enter the common ratio, r.
- Choose how many terms to preview.
- Set a tolerance target for tiny terms.
- Click Test Convergence.
- Read the result card above the form.
- Review the generated table and graph.
- Download the CSV or PDF report if needed.
Frequently Asked Questions
1) What does this calculator test?
It checks whether an infinite geometric series converges or diverges. It also calculates preview terms, partial sums, the infinite sum when valid, and a remainder estimate.
2) When does a geometric series converge?
A geometric series converges only when the absolute value of the common ratio is less than 1. That condition forces the terms to shrink toward zero.
3) Why can a series diverge when r = 1?
When r equals 1, every term stays the same size. The terms never decrease, so the partial sums keep growing and the infinite series does not settle.
4) What happens when r = -1?
The terms flip between positive and negative values without shrinking. The partial sums oscillate and fail to approach one stable limit.
5) Why does the calculator show a partial sum for divergent cases?
A finite partial sum is still meaningful. It shows what the first selected terms add to, even when the corresponding infinite series does not converge.
6) What is the remainder estimate?
The remainder estimate measures how much value is left after the chosen partial sum. It is useful only for convergent geometric series with |r| less than 1.
7) Can the first term be negative?
Yes. The first term may be positive, negative, or zero. The convergence decision depends on the ratio magnitude, not only the sign of the first term.
8) What does the graph display?
The graph plots individual term values and partial sums together. This makes shrinking terms, oscillation, growth, and stabilization easier to compare visually.