Monotonous sequence of numbers

This article deals with the monotony of episodes in . For the monotonic terms of other sequences see monotonic sequence of sets or monotonic sequence of functions . ${\displaystyle \mathbb {R} }$

A monotonous sequence of numbers is a special sequence in which requirements are placed on the growth behavior of the sequence. If the successor members get bigger and bigger, then the sequence is called a monotonically growing sequence or monotonically increasing sequence , if they become smaller and smaller, it is called a monotonically falling sequence . A tightening of the requirements then provides the concept of the strictly monotonically increasing sequence and strictly monotonically decreasing sequence . The monotony of a sequence is an important means of showing the convergence of sequences and can be understood as a special case of a monotonic mapping .

definition

If a sequence of real numbers is given, this sequence is called ${\displaystyle (a_{i})_{i\in \mathbb {N} }=(a_{1},a_{2},a_{3},\dots )}$

• Monotonically increasing or monotonically increasing , if that applies to all . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n+1}\geq a_{n}}$
• Strictly monotonically increasing or strictly monotonically increasing , if that applies to all . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n+1}>a_{n}}$
• Monotonously decreasing when that applies to everyone . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n+1}\leq a_{n}}$
• Strictly decreasing monotonically if that applies to all . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle a_{n+1}<a_{n}}$
• Monotone when they monotonically increasing or monotonically decreasing.
• Strictly monotonic if it is either strictly monotonically increasing or strictly monotonically decreasing.

Examples

• The sequence is neither monotonically increasing nor decreasing, therefore not monotonous either. ${\displaystyle (a_{k})_{k\in \mathbb {N} }=\left((-1)^{k}\right)_{k\in \mathbb {N} }=(-1,1,-1,1,-1,1,\dotsc )}$
• The sequence is strictly monotonically decreasing, because if one forms the difference between two consecutive sequence values , it is always genuinely positive, hence is . This sequence is thus in particular also monotonically decreasing and thus also monotonously. ${\displaystyle (a_{k})_{k\in \mathbb {N} }=\left({\tfrac {1}{k}}\right)_{k\in \mathbb {N} }=(1,{\tfrac {1}{2}},{\tfrac {1}{3}},\dotsc )}$ ${\displaystyle a_{k}-a_{k+1}={\tfrac {1}{k(k+1)}}}$ ${\displaystyle a_{k}>a_{k+1}}$
• The sequence is strictly monotonously increasing. The reasoning works exactly as above, but with the opposite sign. ${\displaystyle (a_{k})_{k\in \mathbb {N} }=\left(-{\tfrac {1}{k}}\right)_{k\in \mathbb {N} }}$
• A sequence that grows monotonically but not strictly monotonically can be defined using the Gaussian bracket as . Since a value is already assumed twice here, the sequence can no longer be strictly monotonic. Nevertheless, it is growing monotonously and therefore also monotonously. ${\displaystyle (a_{k})_{k\in \mathbb {N} }=\left(\lfloor {\tfrac {k}{2}}\rfloor \right)_{k\in \mathbb {N} }=(0,1,1,2,2,3,3,\dotsc )}$

properties

• A sequence is a constant sequence if and only if it is monotonically increasing and monotonically decreasing at the same time.
• Every monotonic sequence converges or diverges in a certain way .
• Every bounded monotonic sequence converges. More precisely, according to the criterion of monotony, a restricted, monotonically falling sequence converges towards the infimum of its subsequent members ; accordingly, a restricted, monotonically growing sequence converges against the supremum of its successors. This also provides the existence of limit values ​​for infinite continued fractions .
• Every sequence has a monotonous subsequence .
• The concept of the monotony of sequences of numbers is a special case of the concept of the monotony of images . To do this, consider the two ordered sets and . Then the sequence is monotonically increasing (decreasing) if and only if the mapping is defined by monotonically increasing (decreasing). ${\displaystyle (\mathbb {N} ,\leq )}$ ${\displaystyle (\mathbb {R} ,\leq )}$ ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$ ${\displaystyle f\colon \mathbb {N} \to \mathbb {R} }$ ${\displaystyle f(k)=a_{k}}$

literature

• Konrad Königsberger : Analysis 1 . Springer, Berlin 2004, ISBN 3-540-41282-4
• Otto Forster : Analysis 1. Differential and integral calculus of a variable. Vieweg, Braunschweig 2004. ISBN 3-528-67224-2