A question about $[c_0,c_1,\ldots,c_n]$ notation for continued fractions I try to understand why by definition   


*

*$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also  

*$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .


Those are continued fractions, and $1$ and $2$ are notes I have in the lecture summery.
But we can add bracket where we we want.  for example:
$[c_0,c_1,\ldots,c_n]=[c_0,c_1,[c_2,\ldots,c_n]]$ 
Thanks!
 A: It's a fairly easy inductive proof that you can write
$$[c_0, c_1, c_2, \dots, c_{n-1}, c_n] = [c_0, [c_1, [c_2, \cdots [c_{n-1}, c_n] \cdots ]]]$$
and you can add or remove any of the sets of square brackets that appear on the right-hand side as you please. That is, you can stick a $[$ where you like as long as the closing $]$ is right at the end, not somewhere in the middle
What you can't do is add square brackets into the left-hand side willy-nilly. For instance, in general,
$$c_0+\dfrac{1}{c_1+\frac{1}{c_2}} = [c_0, [c_1, c_2]] \ne [[c_0, c_1], c_2] = c_0+\dfrac{1}{c_1}+\dfrac{1}{c_2}$$
A: By definition:
$$[c_0,c_1,\ldots,c_n]:=c_0+\cfrac{1}{c_1+\cfrac{1}{c_2+\cfrac{1}{\ddots+\cfrac{1}{c_n}}}}=c_0+\cfrac{1}{[c_1,c_2,\ldots,c_n]}=[c_0,[c_1,\ldots,c_n]]\;,\;etc.$$
So yes: you can put the brackets where you want on the right.
A: Usually one wouldn't write such a thing as $[c_0; [c_1, \ldots, c_n]]$, because elements appearing in a continued fraction representation are supposed to be positive integers (except the one before the semicolon, if there is one, which may be a negative integer).
But it doesn't contradict anything to write this. Well, from the definition of a continued fraction, you have
$$[c_0; c_1, \ldots, c_n] = c_0 + \frac 1 {[c_1, \ldots, c_n]}$$
And also
$$[c_0; c_1] = c_0 + \frac 1 {c_1}$$
Can you see how claim 1 follows from this?
For claim 2, try using induction with claim 1.
Note: I added a semicolon after $c_0$ because the index "$0$" seems to indicate that is what you were looking for.
