How to use the series convergence calculator
Prepare the values for this calculation
Classify convergence for geometric and p-series using their standard convergence tests. Use the formula below to check that it answers your question. Read the input units before transferring figures from a statement, measurement or dataset.
Enter and check each input
- Series family: choose Geometric Σ a·r^n, p-series Σ 1/n^p. Match this selection to the data you enter.
- Geometric first term a: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- Geometric ratio r: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- p-series exponent p: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
The input table gives a complete example. Keep units and time periods consistent across related fields.
Calculate and resolve any input error
Choose Calculate after entering the values. Changes also update the answer after a short pause. If an error appears, follow its message, correct the field and calculate again. Reset restores the example values.
Interpret the answer
Check Convergence first. With the unchanged example inputs, it is Converges. Compare the additional metrics with the primary result; each has its own label and unit.
Compare and keep a useful result
Read the chart’s axis labels and data table. A scenario curve changes one stated input while holding the others fixed; it does not add external observations. Use Download graph to save an available chart as a PNG.
Use Copy result or Download CSV to keep the answer. Share tool shares the page address without placing your input values in the URL. Recheck inputs and assumptions before using the result for a decision.
Series Convergence Calculator formula
A geometric series Σar^n converges iff |r|<1. A p-series Σ1/n^p converges iff p>1.
Inputs and units
| Input | Format / range | Example |
|---|---|---|
| Series family | Select an option | Geometric Σ a·r^n |
| Geometric first term a | Number | 1 |
| Geometric ratio r | Number | 0.5 |
| p-series exponent p | Number | 2 |
Worked example
Using the example inputs above, the convergence is Converges.
- Convergence
- Converges
- Absolute ratio |r|
- 0.5
- Infinite sum
- 2
Results and assumptions
The output uses the convergence convention and formula shown above. Match the entered units and source assumptions when comparing results.
Fix an input or unexpected result
Geometric first term a is required. Enter geometric first term a using the format and bounds in the input table. Zero and an empty field are different: use zero only if it is allowed and represents your data.
If your answer differs from the example, choose Reset and compare every field, including selected units and options. Check percentage versus decimal entry and whether values describe the same period. Round only the final displayed answer when reproducing the formula.
If a discrepancy remains, report the calculator issue with the page address, non-sensitive example inputs, actual result, expected result and a reference for your calculation.