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Yes, a constant sequence (a number repeated indefinitely) is inceed monotonic: it is both monotonic non-decreasing, and monotonic non-increasing. Hence, one can require that a sequence be strictly monotonic increasing or strictly monotonic decreasing.

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## Is every constant sequence monotone?

Yes, every constant sequence is monotone, in fact simultaneously monotone non-decreasing and monotone non-increasing. yes, because constant sequence is both increasing and decreasing sequence. so that it is monotonic.

## Is the sequence a n = 3 monotonic?

A sequence where a n ≥ a n + 1 for all n ∈ N is monotonic. So given that the sequence a n = 3 is all the same numbers and is neither increasing or decreasing, is it monotonic? Yes, a constant sequence (a number repeated indefinitely) is inceed monotonic: it is both monotonic non-decreasing, and monotonic non-increasing.

## How do you know if a sequence is convergent or monotonic?

A sequence may increase for half a million terms, then decrease; such a sequence is not monotonic. If {a n } is both a bounded sequence and a monotonic sequence, we know it is convergent. If {a n } is convergent, though, it may or may not be monotonic.

## Why can’t a monotonic function have more than one constant?

Since a monotonic function has some values that are constant in its domain, this means that there would be more than one value in the range that maps to this constant value.

## Monotonic Sequences and Bounded Sequences – Calculus 2

## More about Is constant sequence monotonic?

### 1. Monotonic Sequence, Series (Monotone): Definition

Nov 13, 2020 · A monotonic (monotone) sequence or monotone series, is always either steadily increasing or steadily decreasing. More formally, a series {a n } is monotonic if either: a i + 1 ≥ 1 for every i ≥ 1. a i + 1 ≤ 1 for every i ≥ 1. If the first is true, the series is monotonically increasing. If the second is true, it is monotonically decreasing.

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### 2. Constant Sequence, Eventually Constant – Calculus How To

Apr 04, 2021 · The constant sequence is a special case of the eventually constant sequence, where a natural number N exists, so that if n ≥ N then a n = a N. In other words, somewhere along the line, the sequence will converge on a natural number. For example, the sequence {1, 2, 3, 4, 4, 4, 4, 4, …} is eventually constant. Constant Sequence Rule. The constant sequence rule …

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### 3. Monotonic function – Wikipedia

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. A function is called monotonically increasing (also increasing or non-decreasin…

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### 4. Monotonous sequences – sangakoo.com

A constant sequence has the form a n = c where c is any number. Any constant sequence has this form. This classification must not be taken as generic since any given sequence is not always increasing or decreasing. If a sequence is increasing or decreasing we will say that it …

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### 5. Monotonic sequences – Sequences and Series

Feb 25, 2018 · Definition. A sequence is called a monotonic sequence if it is increasing, strictly increasing, decreasing, or strictly decreasing, Examples. The following are all monotonic sequences: The sequence (as it is strictly increasing). The sequence (as it is increasing — and decreasing!). The sequence (as it is decreasing).

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### 6. Monotonic Sequences – Bridgewater State University

Prove that the sequence which is defined recursively by \begin{align*} s_1 = 1 \amp\amp s_{n+1}=\sqrt{1+s_n} \end{align*} converges. Proof: The sequence is monotonic: If this is indeed true, then all its terms will follow the pattern suggested by its first two, namely that \begin{align*} s_1 = 1 \amp \amp \lt \amp \amp \sqrt{2} = s_2. \end{align*}

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### 7. How can a constant function be called monotonic? Also, is it

A constant function can’t be strictly monotonic, but just “monotonic” isn’t so fussy. A function is both monotonically increasing and monotonically decreasing over any …

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### 8. Determining whether a series is increasing, decreasing, or not …

Jun 01, 2021 · Sequences are always either monotonic or not 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.

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### 9. calculus – Determine if the following sequence is monotonic …

Sep 15, 2019 · We now note the the denominator is always positive (as sum of two square roots), so the sign of x n − x n + 1 will be determined by the numerator: 2 x n − 1 − 2 x n, but using our IH we know it’s positive so x n − x n + 1 > 0, and we proved by induction that the sequence is monotonically decreasing.

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