Calculator
Formula Used
The calculator studies the series Σ a(n) by testing the absolute series Σ |a(n)|.
Ratio test: L = lim |a(n + 1)| / |a(n)|. The absolute series converges when L < 1 and diverges when L > 1.
Root test: L = lim |a(n)|1/n. The absolute series converges when L < 1 and diverges when L > 1.
Power estimate: p ≈ log(|a(n)| / |a(2n)|) / log(2). A value above 1 suggests p-series style convergence.
How to Use This Calculator
- Enter the general term of the series with n as the index.
- Choose the starting value of n and the number of terms.
- Set a large n value for numerical ratio and root estimates.
- Press Calculate to show the decision above the form.
- Review the term table, then export the report as CSV or PDF.
Example Data Table
| Series term | Suggested input | Expected evidence | Likely result |
|---|---|---|---|
| (-1)n / n2 | ((-1)^n)/(n^2) | p estimate near 2 | Absolutely convergent |
| (-1)n / n | ((-1)^n)/n | p estimate near 1 | Not absolutely convergent |
| 1 / n1.5 | 1/(n^1.5) | p estimate near 1.5 | Absolutely convergent |
| 1 / 2n | 1/(2^n) | Ratio near 0.5 | Absolutely convergent |
Understanding Absolute Convergence
An infinite series may add positive and negative terms. That can make the visible sum look stable. Absolute convergence checks a stronger question. It asks whether the series formed from absolute values also settles to a finite number. When that happens, rearranging terms cannot change the final sum. This is why the test is important in statistics, numerical analysis, probability, and approximation work.
Why This Calculator Helps
This calculator accepts a general term written with n. It evaluates the original term and the absolute term across many values. It then estimates several common tests. The ratio test compares neighboring absolute terms. The root test checks the nth root of a large absolute term. A power estimate compares decay against a p series. Partial sums show how fast the accumulated absolute values are moving.
Reading The Result
A result below one in the ratio or root test supports absolute convergence. A result above one supports divergence. A result close to one is inconclusive. That does not mean the series fails. It means another test may be needed. Many important series, such as p series, return one under ratio and root checks. For those cases, the decay estimate gives helpful guidance.
Good Input Practice
Use standard operators such as +, -, *, /, and ^. Functions like sin, cos, exp, log, sqrt, and abs are supported. Start with a simple expression first. Increase the term count when partial sums change slowly. Increase the large n value when the test result seems unstable. Very oscillatory expressions may need manual review.
Limits Of Numeric Testing
No numeric calculator can prove every series automatically. Rounding, overflow, cancellation, and slow decay can hide the true behavior. Treat the decision as strong evidence, not a formal proof. Use the formula notes and the table to explain your reasoning. For class work, combine the output with a written comparison, ratio, root, alternating, or integral test proof.
In statistical series, absolute convergence also supports stable expectation calculations. It helps when summing probabilities, errors, weights, or transforms. A stable absolute sum reduces order sensitivity and improves confidence in computed models. It also clarifies whether conditional signs are masking poor magnitude decay in long calculations and simulations during repeated experiments and audits today.
FAQs
What is absolute convergence?
Absolute convergence means the series made from absolute values, Σ |a(n)|, converges. If that happens, the original series also converges.
What does conditional convergence mean?
Conditional convergence means Σ a(n) converges, but Σ |a(n)| diverges. Alternating harmonic behavior is a common example.
Why can the ratio test be inconclusive?
The ratio test is inconclusive when its limit is near one. Many p series have this behavior, so another test is needed.
Why can the root test be inconclusive?
The root test is inconclusive when the nth root estimate is near one. It cannot separate many slowly decaying series.
Can this calculator prove convergence?
It gives numerical evidence and test guidance. A formal proof should still use exact limits, comparisons, or other analytic arguments.
Which functions can I type?
You can use abs, sqrt, log, log10, exp, pow, sin, cos, tan, and basic arithmetic with n.
What if my expression gives an error?
Check parentheses, unsupported words, division by zero, and very large values. Try a simpler expression before adding functions.
Why does the partial sum keep growing?
A growing absolute partial sum often suggests divergence, especially when terms decay slowly. Use ratio, root, and comparison evidence together.