This discussion centers on the definition and examples of monotonic sequences, specifically addressing the question of whether the sum of two monotonic sequences can be non-monotonic. Participants propose examples such as the sequences {1/x} and {-1/x}, which sum to a constant value, thus demonstrating a non-monotonic result. The conversation also highlights the distinction between monotonic and strictly monotonic sequences, referencing various mathematical sources including Wikipedia and Rudin's "Principles of Mathematical Analysis". Ultimately, the consensus is that a constant sequence is considered monotonic, despite differing interpretations of the term.
PREREQUISITESMathematics students, educators, and anyone interested in the properties of sequences and their applications in calculus and analysis.
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))."