Sequence Bounded Calculator

Example Data Table

Formula Used

A sequence is bounded above when there is a number M such that a_n ≤ M for every checked index. It is bounded below when there is a number m such that m ≤ a_n. It is bounded when both conditions hold.

For the sampled range, the calculator uses m = min(a_n) and M = max(a_n). It also checks candidate bounds using m ≤ a_n ≤ M with your tolerance. Tail mean uses the average of the last selected terms.

Arithmetic rule: a_n = a₁ + (n - 1)d. A nonconstant arithmetic sequence is unbounded. Geometric rule: a_n = a₁r^n-1. It is bounded when |r| ≤ 1, or when the first term is zero.

How to Use This Calculator

1. Select formula, list, arithmetic, or geometric mode.
2. Enter your rule, list values, or common sequence parameters.
3. Choose the start index, term count, tolerance, and tail window.
4. Add candidate bounds when you want a direct bound test.
5. Press the calculate button and review the result above the form.
6. Download the CSV or PDF report when needed.

Bounded Sequences in Practice

Why Boundedness Matters

A bounded sequence stays inside fixed lower and upper limits. This idea appears in calculus, analysis, numerical methods, and proof writing. Students often test many terms first. Then they use a theorem to confirm the pattern. This calculator supports that first exploration. It also records each checked term for review.

Input Options

The tool accepts a direct formula, a pasted list, an arithmetic rule, or a geometric rule. Formula mode uses n as the index. List mode treats supplied values as a finite sequence. Arithmetic and geometric modes add helpful rule based checks. These checks can identify clear unbounded cases. They can also recognize common bounded patterns.

Upper and Lower Bounds

A sequence is bounded above when every term is less than or equal to one number. It is bounded below when every term is greater than or equal to one number. It is bounded when both statements hold. For a finite sample, the smallest and largest checked terms always give sample bounds. For an infinite sequence, a formal proof may still be needed.

Monotonic and Tail Checks

The calculator also examines monotonic behavior. It compares nearby terms with your chosen tolerance. It reports increasing, decreasing, constant, or mixed behavior. This helps when using convergence theorems. A monotone sequence with a true bound must converge. A nonmonotone sequence may still be bounded, but it needs different reasoning.

Tail analysis estimates long run behavior. The final terms are averaged. The maximum tail deviation is reported. Small deviations may suggest a limit. Large deviations may show oscillation or growth. This estimate is not a proof, but it is useful.

Best Workflow

Use more terms when a sequence changes slowly. Use a smaller tolerance for precise decimal work. Enter candidate bounds when your assignment gives proposed values. The calculator checks whether every sampled term respects them. Export the table when you need evidence in notes, reports, or worksheets.

Advanced users can compare several rules by changing the formula. Try expressions like 1/n, (-1)^n, n/(n+1), or sin(n)/n. Watch how bounds, tails, and monotonic status change. These experiments build intuition before writing a rigorous solution.

The example table shows typical inputs. It gives a quick benchmark today. You can replace those values with your own sequence conditions and assignment targets too.

FAQs