Study limits, bounds, and oscillation with sequence controls. Compare families, inspect terms, and spot divergence. Clear outputs help verify upper and lower bounds accurately.
This example uses the shifted reciprocal sequence aₙ = 2 + 5 / (n + 1).
| n | Formula result | Interpretation |
|---|---|---|
| 1 | 4.500000 | Largest term in this family for n ≥ 1 |
| 2 | 3.666667 | Still above the limit 2 |
| 3 | 3.250000 | Sequence decreases |
| 4 | 3.000000 | Approaches the horizontal limit |
| 5 | 2.833333 | Bounded between 2 and 4.5 |
| 6 | 2.714286 | Confirms bounded, convergent behavior |
aₙ = a₁ + (n - 1)d. It is bounded only when the common difference is zero, or one-sided bounded when the difference has a fixed sign.
aₙ = a₁ · r⁽ⁿ⁻¹⁾. It is bounded for |r| ≤ 1 when the sequence does not grow in magnitude. It becomes unbounded when magnitude increases without limit.
aₙ = c + a / (n + b). For b > -1, the denominator stays positive for all n ≥ 1, so the sequence stays between its first term and the limit c.
aₙ = c + a · (-1)ⁿ / (n + b). The oscillation shrinks in magnitude and the sequence converges to c, while remaining inside a fixed interval.
aₙ = (p·n + q) / (r·n + s). When the denominator never becomes zero at a positive integer, the sequence is bounded and converges to p / r if r ≠ 0.
The calculator also reports sampled minimum, sampled maximum, average value, sign changes, and monotonicity across your chosen term window.
A sequence is bounded if every term stays inside some fixed interval. Bounded above means terms never exceed one constant. Bounded below means terms never fall below one constant.
No. A finite list is always bounded because it has a smallest and largest value. That does not prove future unseen terms remain inside the same interval.
Yes. The classic example is an alternating pattern such as (-1)ⁿ. Its terms stay within fixed limits, but they do not settle to one value.
Sampled bounds come from the displayed term window. Theoretical bounds use known formulas for built-in families. Both are useful because samples show behavior visually while theory explains the full sequence.
A maximum is an actual largest term achieved by the sequence. A supremum is the least upper bound, which may exist even when no term reaches it exactly.
That condition keeps n + b positive for all positive integers n. It avoids denominator problems and makes the bound interpretation cleaner and mathematically reliable.
It measures how often nonzero terms switch between positive and negative values in the displayed sample. This helps identify oscillation, alternating behavior, or movement across the horizontal axis.
Increase the sample size when a sequence changes slowly, oscillates, or approaches a limit gradually. A longer window gives a better visual sense of long-run behavior.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.