Im trying to learn some concepts by reading notation, but I need to know if I understand it correctly.
I read that one could use the index notation $f_n$ instead of $f(n)$, is that right? Wikipage say that $f_n$ is called the n'th element of the sequence in $f(n)$ (typically when the the domain is the set of the natural numbers).
But $n$ in $f(n)$ doesnt necessarily mean the same? It's a placeholder for a number in a function like for example:
$$f(n) = 2^n$$
And
$$f_n = 2^n$$
just does not seem right to me. But is it valid and/or common?
If we define $f^n$ as the n'th iterate, then I can agree that in a sequence $n$ is the n'th element. So from the above statement taken from the wikipage, is the n'th element the same as the n'th iterate? But both of them are defined differently: $f^n$ and $f_n$?
However I am confused by the notation, because there are three ways to note:
$f^n$, $f_n$ and $f(n)$.
Please, I am interested in learning more about Iterated Function Theory and know the difference between pointwise convergence/local uniform convergence and uniform convergence. But before that, I need to understand this basic theory of functions and notation.
Since I am interested in sequences in iterated functions, I want to be able to describe how n'th element, m'th iterate and so on is notated.
I have not found one good example online that explains their differences concretely. Thanks.
Best Answer
$f^n$ is usually used to mean the $n$th power of $f,$ rather than the $n$th iterate:$$f^n(x)= \overbrace{f(x)f(x)\dots f(x)}^{n \text{ factors}}$$ When it means the $n$th iterate, the author will typically say so.
You can usually count on $f(n)$'s meaning the value of the function $f$ at $n$. $f_n$ could mean $f(n),$ particularly when the domain is the natural numbers, but more usually it means the $n$th function in a sequence of functions.
Still, there aren't any hard-and-fast rules. You have to determine the meaning from the context -- just like reading anything else.