Formula Used
Ratio test: L = lim |a(n+1)/a(n)|. If L < 1, the series converges absolutely. If L > 1, it diverges. If L = 1, the test is inconclusive.
Root test: L = lim nth-root(|a(n)|). The same decision rules apply as the ratio test.
Geometric test: Σ ar^n converges when |r| < 1. Its sum is a/(1-r).
P-series test: Σ 1/n^p converges when p > 1. It diverges when p ≤ 1.
Alternating test: Σ(-1)^n b(n) converges when b(n) decreases eventually and approaches zero.
Limit comparison: If 0 < L < ∞, both positive series share the same behavior.
How to Use This Calculator
- Select the test that matches your known series form.
- Enter the value needed by that test.
- Add optional sample terms for a partial sum check.
- Press Calculate to show the result above the form.
- Use CSV or PDF buttons to download the report.
Example Data Table
| Series Type |
Given Value |
Best Test |
Decision |
| Σ 3(1/2)^n |
|r| = 0.5 |
Geometric |
Convergent |
| Σ 1/n^2 |
p = 2 |
P-Series |
Convergent |
| Σ 1/n |
p = 1 |
P-Series |
Divergent |
| Σ n!/n^n |
L = 1/e |
Ratio Test |
Convergent |
| Σ (-1)^n/n |
b(n) decreases to 0 |
Alternating |
Conditionally convergent |
Understanding Series Convergence
A series adds infinitely many terms. The question is simple. Does the running total approach a fixed number? If it does, the series converges. If it does not, the series diverges. A calculator helps because different series need different tests.
Why Tests Matter
No single test solves every case. A geometric series depends on the common ratio. A p-series depends on the power. Ratio and root tests work well for factorials, powers, and exponential terms. The term test catches an immediate failure. Alternating tests handle signs that switch.
How This Tool Helps
This calculator lets you choose the best known pattern. You can enter the limit, exponent, ratio, or comparison value. Then it returns the decision, the rule used, and short reasoning. It also supports report downloads. This makes your work easier to review.
Interpreting Results
A convergent result means the infinite sum has a finite value. Absolute convergence is stronger. It means the series still converges after taking absolute values. Conditional convergence means signs are essential. Divergence means the partial sums do not settle.
Using Caution
Some tests can be inconclusive. This does not mean the series converges or diverges. It only means that test cannot decide. When this happens, try another method. For example, a ratio test limit of one often needs comparison, integral, or alternating analysis.
Practical Study Notes
Always start with the nth-term test. If terms do not approach zero, stop. Next, check for simple geometric or p-series forms. Then try ratio or root tests for expressions with powers or factorials. For positive series, comparison tests are powerful. For alternating signs, check decreasing positive terms and a zero limit.
Exporting And Checking Work
Saved results are useful for class notes and grading records. A CSV file opens in spreadsheets. A PDF report is easier to share. Keep the entered values beside the decision. This prevents confusion later. It also shows which theorem supported the answer.
Common Mistakes
Do not use the alternating test on unsigned series. Do not claim convergence only because terms become small. They must approach zero, and more proof is usually needed. Also remember endpoints of power series need separate tests. The radius alone does not finish that work carefully.
FAQs
What does convergence mean?
Convergence means the infinite sum approaches a finite number. The partial sums settle closer to one value as more terms are added.
Which test should I try first?
Start with the nth-term test. If terms do not approach zero, the series diverges. Then check for geometric, p-series, ratio, root, comparison, or alternating patterns.
Why can a test be inconclusive?
A test is inconclusive when its rule cannot decide the series. It does not prove convergence or divergence. Another test is needed.
What is absolute convergence?
Absolute convergence means the series of absolute values converges. It is stronger than ordinary convergence and often follows from ratio or root tests.
What is conditional convergence?
Conditional convergence means the signed series converges, but the absolute value series diverges. Alternating harmonic series are a common example.
Can this calculator solve every symbolic series?
No. It uses entered limits, powers, ratios, and known comparison behavior. It guides the decision when those values are already known.
How does the PDF option work?
The PDF button creates a simple report. It includes the selected test, decision, formula, reasoning steps, and extra entered details.
Does the partial sum prove convergence?
No. A partial sum is only a numerical check. Infinite convergence needs a valid theorem, such as ratio, root, comparison, or p-series test.