Calculator Inputs
This tool studies series in the form
Σ ± 1 / (n + a)^p.
It compares signed convergence with absolute convergence.
Example Data Table
Use these examples to compare conditional, absolute, and divergent behavior.
| p Value | Series Type | Signed Result | Absolute Result | Classification |
|---|---|---|---|---|
| 0.5 | Alternating p-series | Converges | Diverges | Conditionally Convergent |
| 1 | Alternating harmonic series | Converges | Diverges | Conditionally Convergent |
| 1.5 | Alternating p-series | Converges | Converges | Absolutely Convergent |
| 0 | Alternating constant terms | Diverges | Diverges | Divergent |
Formula Used
The calculator studies this alternating shifted power series:
Σ (-1)^k / (n + a)^p
The matching absolute series is:
Σ |(-1)^k / (n + a)^p| = Σ 1 / (n + a)^p
If p > 1, the absolute series converges.
The original series is absolutely convergent.
If 0 < p ≤ 1, the alternating series converges.
Its absolute series diverges.
The series is conditionally convergent.
If p ≤ 0, the terms do not approach zero.
The series diverges by the term test.
For alternating convergence, the error after N terms is bounded by the next omitted positive term:
Error ≤ 1 / (N + a)^p.
How to Use This Calculator
- Enter the power value p.
- Enter an optional offset a.
- Choose the starting value of n.
- Select how many terms should be used in the partial sum.
- Choose whether the first term is positive or negative.
- Enter a tolerance value for the next omitted term check.
- Press the calculate button.
- Review the classification, partial sums, and test logic.
- Use CSV or PDF buttons to save the result.
Understanding Conditional Convergence
A Middle Case
A conditionally convergent series sits between two common outcomes. Its signed series settles toward a finite value, yet the series made from absolute values does not. This happens because positive and negative terms cancel in a controlled pattern. The classic example is the alternating harmonic series. It converges, but its absolute form becomes the harmonic series, which diverges.
Why the Test Matters
This calculator focuses on alternating p series, because they show the idea clearly. The positive term is one divided by a shifted power. When p is greater than zero, the terms move toward zero. They also shrink as n grows. That lets the alternating series test confirm convergence. The absolute series follows the p series rule. It converges only when p is greater than one.
Reading the Result
A result marked conditional means the signs create enough cancellation, while the absolute values remain too large overall. A result marked absolute means the series is stronger. It still converges after every sign is removed. A divergent result means the term test or p series rule fails. In that case, partial sums may look active, but they do not prove convergence.
Using Partial Sums
Partial sums give a numerical preview. They are not a complete proof by themselves. More terms usually improve the estimate for alternating series. The next omitted term gives a simple error bound when the alternating test applies. For example, after N terms, the error is no larger than the next positive term.
Important Limits
Conditional convergence is delicate. Rearranging the terms can change the final sum. This is why absolute convergence is safer in many applications. Use the classification, the absolute check, and the error estimate together. They give a stronger view than a single decimal output. The tool is useful for homework checks, study notes, numerical experiments, and quick reviews of infinite series behavior.
Best Practices
Start with symbolic reasoning before trusting decimals. Choose enough terms for a stable preview. Keep the offset valid so denominators stay positive. Compare the signed result with the absolute result. Save exports when you need proof steps, examples, or repeated classroom calculations with clear written notes.
Frequently Asked Questions
What is conditional convergence?
Conditional convergence means a signed infinite series converges, but the related absolute value series diverges. The signs create cancellation that makes the original series settle.
What is absolute convergence?
Absolute convergence means the series still converges after every term is changed to its absolute value. This is stronger than conditional convergence.
Why is the alternating harmonic series conditional?
The alternating harmonic series converges by the alternating series test. Its absolute version is the harmonic series, which diverges.
What does the p value control?
The p value controls how fast terms shrink. Larger p values make terms smaller faster. For absolute p-series convergence, p must be greater than one.
Can partial sums prove convergence?
Partial sums help estimate behavior, but they do not prove convergence alone. Use them with the correct convergence test and error bound.
What happens when p is greater than one?
The absolute series converges. Since absolute convergence is stronger, the original alternating series also converges absolutely.
What happens when p is between zero and one?
The alternating series converges, but the absolute p-series diverges. That creates a conditionally convergent result.
Why does rearranging terms matter?
Conditionally convergent series are sensitive to order. Rearranging terms can change the sum or break convergence behavior.