What is monotonic and bounded sequence?

What is monotonic and bounded sequence?

In this section, we will be talking about monotonic and bounded 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.

What is meant by monotonic sequence?

Monotone Sequences. Definition : We say that a sequence (xn) is increasing if xn ≤ xn+1 for all n and strictly increasing if xn < xn+1 for all n. Similarly, we define decreasing and strictly decreasing sequences. Sequences which are either increasing or decreasing are called monotone.

How do you know if a sequence is monotonic?

If a sequence is monotonic, it means that it’s always increasing or always decreasing. If a sequence is sometimes increasing and sometimes decreasing and therefore doesn’t have a consistent direction, it means that the sequence is not monotonic.

How do you tell if a sequence is increasing or decreasing or monotonic?

Definition 6.16. Monotonic Sequence.

1. If an
2. If an≤an+1 a n ≤ a n + 1 for all n, then the sequence is non-decreasing .
3. If an>an+1 a n > a n + 1 for all n, then the sequence is decreasing or strictly decreasing .

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’.

What is bounded and unbounded sequence?

A sequence an is bounded below if there exists a real number M such that. M≤an. for all positive integers n. 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.

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.

What is unbounded sequence?

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.

What is the monotonic sequence theorem?

Monotone Sequence Theorem: (sn) is increasing and bounded above, then (sn) converges. Intuitively: If (sn) is increasing and has a ceiling, then there’s no way it cannot converge.

How can I prove that this sequence is monotonic?

– show the differences in consecutive terms have the same sign (or is zero) – show derivative of associated function has same sign (or is zero) – Given the sequence is always strictly positive or always strictly negative, show the absolute ratio between consecutive terms is always weakly greater than or always weakly less than 1

How to show that this sequence is monotonic?

– Prove that an < 2 for all n ∈ N. – Prove that {an} is an increasing sequence. – Prove that limn→∞an = 2.

What is a monotone sequence?

If {an} is increasing or decreasing, then it is called a monotone sequence. The sequence is called strictly increasing (resp. strictly decreasing) if an < an + 1 for all n ∈ N (resp. an > an + 1 for all n ∈ N. It is easy to show by induction that if {an} is an increasing sequence, then an ≤ am whenever n ≤ m.

What does monotonic mean in calculus?

In calculus and analysis. In calculus, a function defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.