If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function.
In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order. Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y), so it reverses the order.
Yes, and irrelevant. That talks about what is true for strictly increasing or decreasing sequences which was not in question here. The question was about monotonic sequences and there is nothing that requires they be strictly increasing or decreasing.nuclearrape66 said:no...wikipedia says this..."Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y))."
A monotonic sequence is a sequence of numbers that either consistently increases or decreases. In other words, the terms in the sequence are either always getting bigger or always getting smaller.
No, a monotonic sequence cannot have both increasing and decreasing terms. It must be consistently increasing or consistently decreasing.
An example of a non-monotonic sum is a sequence where the terms alternate between increasing and decreasing, such as 1, 3, 2, 4, 3, 5, 4, 6, ...
The main difference is that a monotonic sequence is consistently increasing or decreasing, while a non-monotonic sequence can have terms that both increase and decrease.
Monotonic sequences have many applications in fields such as finance, biology, and engineering. For example, they can be used to model the growth or decline of populations, the stock market, or the spread of diseases.