My Question: Let $\Delta(a_n)$ be defined by $$(a_n)\in \Delta(a_n)\iff \forall n\in\mathbb N: |\epsilon_n| \le |a_n|$$

I guess it is very likely that this notation is already used in mathematical literature. Can you provide a reference for it, please?

Reason for my question: The big O notation $O(a_n)$ has two equivalent definitions (for strictly positive sequences $(a_n)$):

  • $(\epsilon_n) \in O(a_n) \iff \exists C_\forall > 0\,\forall n\in \mathbb N: |\epsilon_n| \le C_\forall |a_n|$
  • $(\epsilon_n) \in O(a_n) \iff \exists C_\infty > 0\,\forall n\in \mathbb N: \limsup_{n\to\infty} \frac{|\epsilon_n|}{|a_n|} \le C_\infty$

Thus one can state the convergence speed with the big O notation but not an estimate for the error (because $C_\forall$ and $C_\infty$ are not known). To state $C_\infty$ I want to use the big Psi notation. For $C_\forall$ I want to use $\Delta(\cdot)$ because $$(\epsilon_n)\in\Delta(C_\forall a_n) \iff \forall n\in\mathbb N: |\epsilon_n| \le C_\forall |a_n|$$ Now I am interested wether this notation is already used.

  • 1
    $\begingroup$ Those are not equivalent, since $a_n$ could be $0$ for infinitely many $n.$ $\endgroup$ – zhw. Nov 13 '15 at 19:58
  • $\begingroup$ I forgot to state that $(a_n)$ shall be a strictly positive sequence... I added it. Thanks! $\endgroup$ – Stephan Kulla Nov 13 '15 at 20:00

Something like this is in common use in real analysis, where for two functions $f,g$ we have

$$ f \leq g \iff f(x) \leq g(x) \text{ for all } x, $$

and it is common to see things like $|f| \leq |g|$ to mean that $|f(x)| \leq |g(x)|$ for all $x$.

This extends naturally to sequences:

$$ |\epsilon| \leq |a| \iff |\epsilon_n| \leq |a_n| \text{ for all } n. $$


Donald Knuth refers to it in "Teach Calculus with Big O" as the A notation:

I would begin my ideal calculus course by introducing a simpler “A notation”, which means “absolutely at most”. For example, A(2) stands for a quantity whose absolute value is less than or equal to 2.

Source: Donald Knuth (June–July 1998). "Teach Calculus with Big O". Notices of the American Mathematical Society 45 (6): page 687.


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