Calculator Inputs
Enter the nth-term formula using n. Use explicit multiplication such as 3*n, not 3n. Supported functions: sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, pow, floor, ceil, round.
Example Data Table
These sample sequences show how the calculator can classify common patterns over a tested range.
| Formula | Typical behavior | Common limit | Expected result |
|---|---|---|---|
| 1/n | Positive and decreasing | 0 | Convergent |
| (-1)^n / n | Alternating with shrinking size | 0 | Convergent |
| (-1)^n | Alternating without shrinking | None | Divergent |
| 2^n | Rapid positive growth | None | Divergent |
| (3*n+1)/(n+2) | Approaches a finite value | 3 | Convergent |
Formula Used
1) Generated terms
The calculator evaluates the entered rule as an for each tested index n. It then studies the sampled values numerically.
2) Tail average
A tail average is computed from the last k tested terms:
Tail average = (am-k+1 + ... + am) / k
3) Tail max deviation
The tool measures how tightly tail terms gather around the tail average:
Max deviation = max |an - tail average|
4) Differences and ratios
It also checks:
Difference = an - an-1
Ratio = an / an-1 when the previous term is nonzero.
5) Numerical classification logic
A sequence is marked likely convergent when recent terms cluster near a stable value or when alternating magnitudes shrink toward zero. It is marked likely divergent when sampled terms grow past the threshold or oscillate without shrinking.
How to Use This Calculator
- Enter the nth-term formula using n and explicit multiplication.
- Choose the starting index and the number of terms to test.
- Set a tolerance to control how close tail terms must be.
- Choose a tail window to define the recent terms used for judgment.
- Optionally enter an expected limit if you already know the target value.
- Click Analyze Sequence to view the result above the form.
- Study the summary cards, the term table, and the Plotly graph.
- Export the report with the CSV or PDF buttons when needed.
FAQs
1) Does this tool prove convergence formally?
No. It gives numerical evidence from sampled terms. A formal proof may still require algebra, comparison tests, squeeze arguments, monotone convergence, or another analytic method.
2) What formula format should I use?
Use n as the variable and write explicit multiplication. For example, enter (3*n+1)/(n+2) instead of (3n+1)/(n+2).
3) Why can a convergent sequence show “inconclusive”?
Some sequences converge slowly. If you test too few terms or use a very strict tolerance, the sampled tail may not look stable enough yet.
4) What does the tail window control?
It sets how many recent terms are used for the numerical judgment. A larger window gives broader evidence, while a smaller window reacts faster to recent behavior.
5) When should I enter an expected limit?
Use it when you know the likely limit from theory or simplification. The calculator then checks how closely the tail terms match that target.
6) Can this calculator handle alternating sequences?
Yes. It checks sign changes and whether absolute values shrink. That helps distinguish sequences like (-1)^n from (-1)^n/n.
7) What does the divergence threshold do?
It sets a practical ceiling for tested magnitudes. When sampled terms exceed that level, the tool treats the behavior as unbounded on the tested range.
8) Which functions are supported?
The calculator supports sin, cos, tan, asin, acos, atan, sqrt, abs, exp, ln, log, pow, floor, ceil, and round, along with constants pi and e.