Calculator Inputs
Formula Used
This calculator studies positive-term series using a supported asymptotic model:
bn = D / [(En + F)r(ln(En + F))s]
Direct comparison rule: If 0 ≤ an ≤ bn eventually and Σbn converges, then Σan converges. If an ≥ bn ≥ 0 eventually and Σbn diverges, then Σan diverges.
Limit comparison rule: If limn→∞ an/bn = L with 0 < L < ∞, then both series have the same behavior. This page also uses the useful consequences from ratio limits 0 and ∞ when the comparison result supports a final decision.
How to Use This Calculator
- Choose whether you want direct comparison, limit comparison, or both.
- Enter the target series parameters A, B, C, p, and q.
- Enter a known comparison series using the same supported format.
- Select the eventual inequality if you know it for large n.
- Set the starting n, the number of sample rows, and display precision.
- Press the calculate button to see the conclusion, steps, table, and graph.
- Use the CSV and PDF buttons to export the result area.
Example Data Table
| Example | Target series | Comparison series | Best test choice | Expected outcome |
|---|---|---|---|---|
| Example 1 | 1 / n2 | 1 / n2 | Limit comparison | Converges |
| Example 2 | 1 / [n(ln n)2] | 1 / [n(ln n)2] | Limit comparison | Converges |
| Example 3 | 1 / n0.5 | 1 / n | Direct or limit comparison | Diverges |
FAQs
1. What does this calculator decide?
It decides whether a positive series converges, diverges, or stays inconclusive under the entered comparison setup. It also shows the reasoning steps, sample values, and a graph.
2. Which series forms are supported here?
This page supports series terms built from linear expressions raised to a power, with an optional logarithmic power. That covers many common p-series and logarithmic comparison problems.
3. Why do I need positive terms?
The direct comparison test and the usual limit comparison setup require nonnegative terms eventually. If your series changes sign, this calculator is not the right tool.
4. When should I choose direct comparison?
Choose direct comparison when you already know an eventual inequality such as aₙ ≤ bₙ or aₙ ≥ bₙ. That gives a quick proof when the comparison series is already classified.
5. When should I choose limit comparison?
Choose limit comparison when the target and comparison look asymptotically similar. It is especially helpful when the inequality is messy but the ratio limit is easy to evaluate.
6. Why might the result be inconclusive?
An inconclusive result means the entered comparison series or inequality does not force a final answer. Try a different benchmark series or switch from direct to limit comparison.
7. What does the ratio limit mean here?
The ratio limit compares the long-run size of the target term to the comparison term. A positive finite limit means both series behave the same way.
8. What do the CSV and PDF buttons export?
The CSV button exports the sample term table. The PDF button captures the complete result area, including the steps, breakdown cards, table, and graph.