Menu Close

Can a sequence be non decreasing and non increasing?

Can a sequence be non decreasing and non increasing?

A sequence which is either increasing, decreasing, non-increasing, or non-decreasing is called a monotone sequence.

Is non decreasing increasing?

2 Answers. Nondecreasing means that the values could stay the same – they don’t decrease but they could increase or stay the same. The values 1, 1, 1, 2 are in nondecreasing order but 1, 2, 3, 4 are increasing.

What is a monotone sequence?

If {an} is increasing or decreasing, then it is called a monotone sequence. Let {an} be a sequence of real numbers. The following hold: If {an} is increasing and bounded above, then it is convergent. If {an} is decreasing and bounded below, then it is convergent.

How do you determine if a sequence is increasing or decreasing?

We call the sequence increasing if ansequence 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.

How do you prove a function is non decreasing?

Non-decreasing functions Recall that a function f:A→B is non-decreasing if x≤y implies f(x)≤f(y) for all x, y in A.

What is a non increasing sequence?

(mathematics) A sequence, {Sn }, of real numbers that never increases; that is, Sn +1≤ Sn for all n. A sequence of real-valued functions, {ƒn }, defined on the same domain, D, that never increases; that is, ƒn +1(x) ≤ ƒn (x) for all n and for all x in D.

What is mean by non-decreasing order?

Nondecreasing order simply refers to the idea of sorting things where subsequent items are greater than, or equal to the previous item. This is different to increasing order because in an increasing ordered list, each item is greater than the previous one.

What is non-decreasing function?

A function is said to be nondecreasing on an interval if for all , where . Conversely, a function is said to be nonincreasing on an interval if for all with . SEE ALSO: Decreasing Function, Monotone Decreasing, Monotone Increasing, Nonincreasing Function.

What is the difference between increasing function and non decreasing function?

A (strictly) increasing function f is one where x_1 < x_2 \implies f(x_1) < f(x_2). A non-decreasing function f is one where x_1 < x_2 \implies f(x_1) \leq f(x_2). The dual terms are (strictly) decreasing and non-increasing (reverse the direction of the inequalities), respectively.

Is every monotone sequence convergent?

We have already seen the definition of montonic sequences and the fact that in any Archimedean ordered field, every number has a monotonic nondecreasing sequence of rationals converging to it.

How do you know if a sequence is monotone?

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. A bounded monotonic increasing sequence is convergent.

Is every increasing sequence bounded below?

Figure 2.4: Sequences bounded above, below and both. Each increasing sequence (an) is bounded below by a1. Each decreasing sequence (an) is bounded above by a1.

Is every convergent sequence is Cauchy?

Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0, beyond some fixed point, every term of the sequence is within distance ε/2 of s, so any two terms of the sequence are within distance ε of each other.

How do you find if a function is bounded?

If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below.

What does it mean when a function is bounded below?

Functions Bounded Below. Definition: A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Answers is in terms of y-values. Any such number b is called a lower bound of f.

Is Lnx bounded?

For 1≤x<∞, we know lnx can be bounded as following: lnx≤x−1√x.

What is log of infinity?

The natural log function of infinity is usually denoted as loge ∞ and is also referred to as the log function of infinity to the base e. When the variable x takes the value of infinity, it becomes e ∞ = ∞. So, limx -> ∞ex = ∞.

What is log a B?

log A + log B = log AB. This law tells us how to add two logarithms together. Adding log A and log B results in the logarithm of the product of A and B, that is log AB. For example, we can write. log10 5 + log10 4 = log10(5 × 4) = log10 20.

What is E in log?

The number e , sometimes called the natural number, or Euler’s number, is an important mathematical constant approximately equal to 2.71828. When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as ln(x) ⁡ .

How do you convert log to LN?

If you need to convert between logarithms and natural logs, use the following two equations:

  1. log10(x) = ln(x) / ln(10)
  2. ln(x) = log10(x) / log10(e)

Is log 0 possible?

log 0 is undefined. It’s not a real number, because you can never get zero by raising anything to the power of anything else. log 1 = 0 means that the logarithm of 1 is always zero, no matter what the base of the logarithm is. This is because any number raised to 0 equals 1.

Why is log (- 3 not defined?

The logarithm of a negative number is not defined as a negative number is equal to the odd power of a negative number. For X to be negative in the earlier relation, a has to be a negative number and b has to be odd. If a were negative, for most values of X, there wouldn’t be a corresponding value for b.

Why can Ln not be negative?

What is the natural logarithm of a negative number? The natural logarithm function ln(x) is defined only for x>0. So the natural logarithm of a negative number is undefined.

Why does log (- 1 have no solution?

The log function is used to undo raising something to a power. Just log() has a default base value of 10. or, if we take log(100), we get 2 because 10^2 = 100. So, if you think, “what power does 10 have to be raised to, to get -1”, you will get nothing.

What is Ln infinity?

1 Answer. Amory W. The answer is ∞ . The natural log function is strictly increasing, therefore it is always growing albeit slowly.

What is the LN of 0?

What is the natural logarithm of zero? ln(0) = ? The real natural logarithm function ln(x) is defined only for x>0. So the natural logarithm of zero is undefined.

Is Ln 0 infinity?

The ln of 0 is infinity.

What Infinity subtracts infinity?

It is impossible for infinity subtracted from infinity to be equal to one and zero. Using this type of math, we can get infinity minus infinity to equal any real number. Therefore, infinity subtracted from infinity is undefined.