What are bounded and monotonic sequences?
We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below.
What is monotonic sequence and examples?
A sequence is said to be monotone if it is either increasing or decreasing. Example. The sequence n2 : 1, 4, 9, 16, 25, 36, 49, is increasing. The sequence 1/2n : 1/2, 1/4, 1/8, 1/16, 1/32, is decreasing. The sequence (1)n1/n : 1, 1/2, 1/3, 1/4, 1/5, 1/6, is not monotone.
What is bounded sequence with example?
A sequence an is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence. For example, the sequence 1/n is bounded above because 1/n≤1 for all positive integers n. It is also bounded below because 1/n≥0 for all positive integers n.
How do you tell if a sequence is bounded and monotonic?
We call the sequence decreasing if an>an+1 a n > a n + 1 for every n . If {an} is an increasing sequence or {an} is a decreasing sequence we call it monotonic. If there exists a number m such that m≤an m ≤ a n for every n we say the sequence is bounded below.
What is bounded sequence?
A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K’, greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K’.
Is every bounded sequence is monotonic?
Only monotonic sequences can be bounded, because bounded sequences must be either increasing or decreasing, and monotonic sequences are sequences that are always increasing or always decreasing.
What is monotonic sequence?
Definition. A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ N. Remarks. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly.
What is a bounded sequence?
What is the monotonic sequence theorem?
Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.
What makes a sequence monotonic?
We need the following. A sequence (an) is monotonic increasing if an+1≥ an for all n ∈ N. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly.
Are all monotonic sequence bounded?
What is monotonic sequence theorem?
In real analysis, the monotone convergence theorem states that if a sequence increases and is bounded above by a supremum, it will converge to the supremum; similarly, if a sequence decreases and is bounded below by an infimum, it will converge to the infimum.