Result
Test Details
Export Result
| n | an | Partial Sum | |an+1 / an| |
|---|
Calculator Inputs
Use formulas with n, such as 1/n^2, (-1)^(n+1)/n, 0.5^n, or 1/(n*log(n)).
Supported functions include sin, cos, tan, log, log10, sqrt, abs, exp, and pow.
Example Data Table
This example uses the series an = 1 / n². It is a p-series with p = 2.
| Formula | Test | Key Value | Expected Result |
|---|---|---|---|
| 1 / n² | P-Series | p = 2 | Converges |
| 1 / n | P-Series | p = 1 | Diverges |
| 0.5ⁿ | Geometric | |r| = 0.5 | Converges |
| 2ⁿ | Geometric | |r| = 2 | Diverges |
Formula Used
The calculator studies a term formula an. For a series, it estimates SN = a1 + a2 + ... + aN. It also checks the nth term rule. If an does not approach zero, then the series diverges.
Ratio test: L = lim |an+1 / an|. If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, the test is inconclusive.
Root test: L = lim |an|1/n. If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, more proof is needed.
P-series rule: Σ 1 / np converges when p > 1. It diverges when p ≤ 1. Geometric rule: Σ arn converges when |r| < 1. It diverges when |r| ≥ 1.
How to Use This Calculator
Enter the formula for an. Choose whether you want to test an infinite series or a sequence. Select an automatic test or choose a specific method. Set the starting value of n, number of terms, and tolerance. Press the calculate button. The result will appear above the form and below the header section.
Use p value when selecting the p-series test. Use common ratio when selecting the geometric test. Review the test notes before trusting the verdict. Download the CSV file for spreadsheet work. Download the PDF file when you need a clean report.
Understand Convergence Before Using the Answer
A converging or diverging calculator helps you test whether an infinite sequence or series approaches a stable value. It does not replace proof, but it gives strong numerical evidence. This page checks term behavior, ratio trends, root trends, geometric rules, p-series rules, and alternating estimates. It also records the first terms, partial sums, and test notes in one clean result.
Why Convergence Matters
Convergence is important in calculus, finance, physics, computing, and modeling. A convergent series can represent a useful total, such as a distance, error bound, investment stream, or approximation. A divergent series grows without a finite total, or its terms fail to settle toward zero. Knowing the difference protects your calculations from false conclusions.
How the Calculator Thinks
The tool evaluates the formula at many positive integer values of n. It builds a list of terms and partial sums. Then it compares late terms, late ratios, and late roots. When the formula matches a known family, such as a geometric series or p-series, the related rule is applied directly. These checks are combined into a verdict and explanation.
Advanced Options for Better Testing
You can choose an automatic test or focus on one method. Increase the number of terms for smoother estimates. Change the tolerance when values are very small. Use the alternating option when signs change regularly. The calculator also reports whether the nth term appears to approach zero, because that is required for any series to converge.
Use Results Carefully
Some series need symbolic proof. Numerical tests can be fooled by slow behavior, cancellation, or formulas that change after many terms. Treat the verdict as a guide. Use the notes, table, and exported files to support your written solution. For classwork, include the selected test, the key limit, and the final convergence statement.
Example Learning Workflow
Start with a simple known series, then compare it with your own formula. Review the generated terms before trusting the verdict. If early values look unusual, increase the sample size. When results disagree across tests, prefer the strongest valid theorem. Save the CSV file for data review, and use the PDF for reports later.
FAQs
1. What does converging mean?
Converging means the sequence approaches a finite limit, or the infinite series approaches a finite total. The terms and sums settle instead of growing endlessly.
2. What does diverging mean?
Diverging means the sequence or series does not approach a finite value. It may grow, oscillate, or fail a required convergence condition.
3. Can this calculator prove convergence?
It gives strong numerical evidence and applies standard rules. Some problems still need a written symbolic proof, especially slow or unusual series.
4. What is the nth term test?
The nth term test checks whether a series term approaches zero. If the term does not approach zero, the series must diverge.
5. When should I use the ratio test?
Use the ratio test for factorials, exponentials, and products. It compares neighboring terms and works well when ratios settle clearly.
6. When should I use the root test?
Use the root test when terms contain powers involving n. It checks the nth root of the absolute term.
7. Why is the result sometimes inconclusive?
Some tests cannot decide every series. Ratio and root tests are inconclusive when the limit is near one.
8. Can I export my result?
Yes. Use the CSV option for spreadsheet data. Use the PDF option for a simple report with verdict, values, and notes.