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. The number m is sometimes called a lower bound for the sequence.
What does it mean for a sequence to be monotonic?
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 a monotonic sequence example?
Monotonicity: The sequence sn is said to be increasing if sn sn+1 for all n 1, i.e., s1 s2 s3 …. … 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.
Is every sequence monotonic?
It turns out that every sequence of real numbers
has subsequence that is monotone
.
How do you show a sequence is monotone and bounded?
if an ≥ an+1 for all n ∈ N. A sequence is monotone if it is either increasing or decreasing.
and bounded, then it converges
.
Does the sequence 1 n converge or diverge?
n=1 an diverges
. n=1 an converges if and only if (Sn) is bounded above.
What is a monotonic relationship?
A monotonic relationship is a relationship that does one of the following: (1)
as the value of one variable increases
, so does the value of the other variable, OR, (2) as the value of one variable increases, the other variable value decreases.
Are all monotonic sequences convergent?
Every
monotonically increasing sequence which is bounded above
is convergent.
How do you determine if a sequence is convergent or divergent?
If we say that a sequence converges, it means that the
limit of the sequence
exists as n → ∞ ntoinfty n→∞. If the limit of the sequence as n → ∞ ntoinfty n→∞ does not exist, we say that the sequence diverges. A sequence always either converges or diverges, there is no other option.
When a monotonic decreasing sequence is convergent?
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.
Is every bounded sequence is monotonic?
Not all bounded sequences, like (−1)n, converge, but if we knew the bounded sequence was monotone, then this would change. if an ≥ an+1 for all n ∈ N. A sequence is monotone if it is either increasing or decreasing. and bounded, then it converges.
What makes a sequence divergent?
If a sequence does not converge, then it
is said to diverge or to be a divergent sequence. For example, the following sequences all diverge, even though they do not all tend to infinity or minus infinity: 1, 2, 4, 8, 16, 32, …1, 0, 1, 0, 1, 0, …
Which of the following is a monotonically increasing sequence?
A sequence (a
n
) is monotonic increasing if
a
n + 1
≥ a
n
for all n ∈ N
. The sequence is strictly monotonic increasing if we have > in the definition. Monotonic decreasing sequences are defined similarly.
What is monotonic pattern?
A missing data pattern is said to be monotone
if the variables Yj can be ordered such that if Yj is missing then all variables Yk with k>j are also missing
. This occurs, for example, in longitudinal studies with drop-out. If the pattern is not monotone, it is called non-monotone or general.
What is monotonic in research?
The term monotonic relationship is a statistical definition that is used to describe
a scenario in which the size of one variable increases as the other variables also increases
, or where the size of one variable increases as the other variable also decreases.
