When a sequence is bounded in an interval, it means that all the terms in the sequence fall within a specific range or interval of numbers. This interval can be defined by a lower and upper bound.
To check if a sequence is bounded in an interval, you can use the limit test. If the limit of the sequence exists and falls within the given interval, then the sequence is bounded. Another way is to find the maximum and minimum values of the sequence and see if they fall within the given interval.
If a sequence is not bounded in an interval, it means that the terms of the sequence are either increasing or decreasing without limit. This could also mean that the sequence is oscillating between two values without converging to a specific value.
Yes, a sequence can be bounded in multiple intervals. This means that the terms of the sequence fall within different ranges or intervals. For example, a sequence could be bounded in the intervals [0, 1] and [3, 5], meaning all the terms fall between 0 and 1, as well as between 3 and 5.
No, a bounded sequence is not always convergent. A sequence can be bounded but not convergent if it has multiple limit points or if it is oscillating between two values without converging to a specific value. However, a bounded sequence must be convergent if it is also monotonic (either increasing or decreasing).