Calculator Input
Formula Used
Finite geometric series:
Sₙ = a(1 - rⁿ) / (1 - r), when r ≠ 1.
Special finite case:
Sₙ = a × n, when r = 1.
Nth term:
Tₙ = a × rⁿ⁻¹.
Infinite geometric series:
S∞ = a / (1 - r), only when |r| < 1.
The calculator also builds each term and adds running totals. This helps verify the formula result step by step.
How to Use This Calculator
- Enter the first term of the geometric series.
- Enter the common ratio between consecutive terms.
- Enter how many terms should be included.
- Select whether you want finite, infinite, or combined results.
- Choose decimal places for the output.
- Press the calculate button.
- Review the sum, nth term, convergence result, graph, and table.
- Use the CSV or PDF button to save your result.
Example Data Table
| First Term | Ratio | Terms | Finite Sum | Infinite Status |
|---|---|---|---|---|
| 5 | 0.5 | 8 | 9.9609375 | Convergent |
| 3 | 2 | 6 | 189 | Divergent |
| 10 | -0.25 | 5 | 8.0078125 | Convergent |
Geometric Series Guide
What a Geometric Series Means
A geometric series adds terms that follow one repeated multiplier. This multiplier is called the common ratio. If the first term is 5 and the ratio is 2, the terms are 5, 10, 20, and 40. Each term grows from the one before it. If the ratio is between -1 and 1, the terms usually shrink. This makes the series useful for decay, discounts, and fractions.
Why the Sum Matters
The sum helps measure the total value of a repeated pattern. Students use it in algebra and calculus. Investors use it in finance models. Engineers use it for signals and repeated losses. A finite sum stops after a fixed number of terms. An infinite sum keeps going forever. The infinite sum exists only when the absolute value of the ratio is less than one.
Reading the Result
The finite sum gives the exact total for your chosen term count. The nth term shows the final term in that range. The convergence message tells whether the infinite sum is valid. The graph shows how terms and running totals behave. A rising curve often means growth. A flattening cumulative curve often means convergence. An alternating ratio can make values move above and below zero.
Using the Tool for Better Checks
Enter simple values first. Then compare them with your manual work. Change the ratio to see how fast the sum changes. Use a ratio close to one for slow convergence. Use a negative ratio to study alternating series. Download the CSV file for spreadsheets. Download the PDF file for assignments or reports. The term table gives a clear audit trail.
FAQs
1. What is a geometric series?
A geometric series is a sum of terms where each term is found by multiplying the previous term by one fixed ratio.
2. What is the common ratio?
The common ratio is the multiplier between two neighboring terms. Divide any term by the previous term to find it.
3. When does an infinite geometric series converge?
An infinite geometric series converges when the absolute value of the common ratio is less than one.
4. What happens when the ratio equals one?
Every term stays the same. The finite sum is the first term multiplied by the number of terms.
5. Can the common ratio be negative?
Yes. A negative ratio creates alternating positive and negative terms. The calculator still handles the sum correctly.
6. Why does my infinite sum show divergent?
It shows divergent when the common ratio is not between -1 and 1. The endless sum then has no fixed total.
7. What does the graph show?
The graph shows each term and the cumulative sum. It helps you see growth, decay, or alternating behavior.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean printable report.