Calculator Input
Example Data Table
| Input b(n) | Alternating form | Expected sequence behavior | Expected series behavior |
|---|---|---|---|
| 1/n | (-1)^(n+1)/n | Converges to 0 | Converges conditionally |
| 1/sqrt(n) | (-1)^n/sqrt(n) | Converges to 0 | Converges by alternating test |
| n/(n+1) | (-1)^n × n/(n+1) | Diverges by oscillation | Diverges by nth-term test |
| 1/n^2 | (-1)^(n+1)/n^2 | Converges to 0 | Converges absolutely |
Formula Used
The calculator studies an alternating term in this form:
a(n) = (-1)^n b(n) or a(n) = (-1)^(n+1)b(n)
For a sequence, convergence usually needs:
lim n→∞ a(n) = L
For a standard alternating sequence with positive b(n), it converges to zero when:
lim n→∞ b(n) = 0
For an alternating series, the common test is:
Σ (-1)^n b(n) converges when b(n) ≥ 0, b(n) decreases, and lim b(n) = 0.
The estimated alternating series error after N terms is:
|R_N| ≤ b(N+1)
How to Use This Calculator
- Enter the positive part of the term as
b(n). - Select the alternating sign pattern.
- Choose the starting value of
n. - Set how many terms you want to display.
- Set a large value of
nfor the tail check. - Press calculate to view the result above the form.
- Review the graph, partial sums, and table.
- Use CSV or PDF export for records.
Alternating Sequence Convergence Guide
What This Calculator Checks
Alternating sequences change sign from one term to the next. They often look confusing because positive and negative values appear together. This calculator separates the sign pattern from the size part. The size part is called b(n). That makes the test easier to read. If b(n) shrinks toward zero, the alternating sequence usually settles at zero.
Why the Limit Matters
A sequence converges when its terms approach one fixed value. For an alternating sequence, the signs keep switching. So the size must shrink. If the size does not shrink, the terms keep jumping between positive and negative values. Then no single limit exists. This is why the tail check is important.
Series Versus Sequence
A sequence checks individual terms. A series checks the sum of terms. These are related but different ideas. The calculator shows both views. The partial sum column helps you study the series. The term column helps you study the sequence. A sequence can approach zero while the related series still needs a stronger test.
Monotonic Behavior
The alternating series test needs b(n) to decrease after some point. The calculator checks sampled values for decreasing behavior. This is a numerical check, not a full proof. Some expressions only become decreasing later. In that case, increase the starting value or the large tail value.
Using the Graph
The graph shows the alternating terms and partial sums. Term values should move closer to zero when the sequence converges. Partial sums should settle when the series converges. If the graph keeps spreading or jumping, the expression may diverge. Always compare the graph with the table and summary.
Best Practice
Use simple expressions first. Try examples like 1/n, 1/n^2, and n/(n+1). Then test harder formulas. Numerical tools are helpful, but final answers in calculus may need written reasoning. Use the result as guidance for a formal solution.
FAQs
1. What is an alternating sequence?
An alternating sequence has terms that switch signs. One term is positive, and the next is negative. A common form is (-1)^n b(n), where b(n) controls the size.
2. When does an alternating sequence converge?
It usually converges when the size part b(n) approaches zero. If the size does not approach zero, the terms keep oscillating and no single limit exists.
3. Is an alternating sequence the same as an alternating series?
No. A sequence studies individual terms. A series studies the sum of terms. This calculator shows both term behavior and partial sum behavior.
4. What is the alternating series test?
The test says an alternating series converges when b(n) is positive, eventually decreasing, and approaches zero. The calculator checks these conditions numerically.
5. Why does the calculator say inconclusive?
Some expressions need symbolic proof. Numerical samples may not capture later behavior. Try a larger tail value or prove monotonic behavior by algebra.
6. What does the error bound mean?
For a valid alternating series test, the next unused b(n) value bounds the remaining error. Smaller next terms mean more accurate partial sums.
7. Can I use functions in b(n)?
Yes. You can use functions like sqrt(), abs(), sin(), cos(), log(), ln(), exp(), floor(), and ceil(). Use n as the variable.
8. Are the results a formal proof?
No. The calculator gives strong numerical guidance. For coursework, support the result with limit rules, monotonic proof, or a named convergence test.