Example Data Table
| Test |
Input values |
Expected result |
Reason |
| p-Series |
p = 2 |
Convergent |
p is greater than 1. |
| Geometric |
a = 5, r = 0.4 |
Convergent |
|r| is less than 1. |
| Ratio |
L = 0.75 |
Absolutely convergent |
L is less than 1. |
| Nth-Term |
Limit = 0.2 |
Divergent |
Terms do not approach zero. |
| Alternating |
Decreasing yes, zero yes |
Convergent |
Alternating rule conditions pass. |
Formula Used
p-Series: Sum C / n^p converges when p > 1 and diverges when p ≤ 1.
Geometric Series: Sum a r^(n-1) converges when |r| < 1. Its sum is S = a / (1 - r).
Ratio Test: L = lim |a_(n+1) / a_n|. Convergent if L < 1. Divergent if L > 1.
Root Test: L = lim nth root of |a_n|. Convergent if L < 1. Divergent if L > 1.
Nth-Term Test: If lim a_n is not zero, the series diverges.
Limit Comparison: If lim a_n / b_n is positive and finite, both series share convergence behavior.
Alternating Test: If positive terms decrease to zero, the alternating series converges.
How To Use This Calculator
- Select the convergence test that matches your series form.
- Enter only the values needed for that selected test.
- Use p for p-series and integral power tests.
- Use L for ratio, root, and nth-term tests.
- Set comparison details when using comparison tests.
- Choose alternating conditions for alternating series.
- Press the calculate button to view the result above the form.
- Use CSV or PDF download buttons to save the steps.
Understanding Convergence Tests
A convergence test studies an infinite series. It asks whether added terms settle toward a finite value. This calculator keeps the idea practical. It compares the chosen rule with the numbers you enter. Then it writes steps that explain the decision.
Why Steps Matter
Students often know a rule, yet miss the condition. A ratio test needs a limit of absolute term ratios. A root test needs an nth root limit. A p-series test checks the exponent. A geometric test checks the common ratio. Clear steps reduce guessing. They also show why a result may be inconclusive.
Choosing the Right Rule
Start with the term shape. Use the geometric test when powers repeat by a fixed ratio. Use the p-series or integral style test when terms look like one over a power of n. Use the ratio test for factorials, exponentials, and products. Use the root test when n appears in an exponent. Use alternating tests when signs switch and positive terms shrink to zero.
Reading the Result
Convergent means the infinite sum approaches a finite value. Divergent means it fails to do that. Absolute convergence is stronger. It means the series of absolute values also converges. Conditional convergence means the alternating series converges, but the absolute series does not. Inconclusive means the selected test cannot decide. It does not prove convergence or divergence.
Practical Study Notes
Enter careful estimates. A limit near one can change the answer. Use exact values when possible. Treat negative ratios by using absolute value. For comparison tests, pick a known benchmark series. Harmonic series diverge. A p-series converges only when p is greater than one. Geometric series converge only when the absolute ratio is less than one.
Exporting Your Work
The download buttons save the calculation record. The CSV file is useful for spreadsheets. The PDF file is useful for notes. Keep both with homework, audits, or lesson plans. Always review assumptions. Many tests require positive terms, monotone behavior, or a valid known comparison. For classroom use, record the test name, entered limit, rule condition, and final status. That trail helps teachers find mistakes. It also helps learners compare several possible tests before trusting one conclusion during careful later review sessions.
FAQs
What does convergence mean?
Convergence means the infinite sum approaches a finite value. The partial sums settle instead of growing without bound or oscillating without control.
What does divergence mean?
Divergence means the series does not approach a finite sum. It may grow, oscillate, or fail a required condition.
When should I use the ratio test?
Use the ratio test for series with factorials, exponentials, products, or terms that simplify well after forming a_(n+1) divided by a_n.
When should I use the root test?
Use the root test when the term contains powers involving n. It is helpful for expressions raised to the nth power.
What does inconclusive mean?
Inconclusive means the selected test cannot decide. It is not a final proof. Try another test or improve the entered limit.
Can this calculator solve every series?
No. It evaluates common convergence decisions from entered values. Some series need algebra, symbolic limits, or deeper comparison work.
What is absolute convergence?
Absolute convergence means the series made from absolute term values also converges. This is stronger than ordinary convergence.
What is conditional convergence?
Conditional convergence means the original series converges, but the absolute value series diverges. Alternating harmonic series are a common example.