swampwiz said:
How about for π?
https://en.wikipedia.org/wiki/Pi#Continued_fractions
The Wiki article says:
It seems that the first approximation (3) is the result of setting the continuing fraction to 0,
That article doesn't use precise terminology for infinite series, so I see where you get the thought that some infinite string of symbols is assigned to be zero. You are getting that thought from the article's use of term "truncate".
If we have an infinite
series denoted by ##a_0+a_1+a_2+...## this notation (suggesting an infinite string of symbols) does not necessarily represent a number. Likewise, the notation ##\sum_{i=0}^n a_i## does not necessarily represent a number. We consider the
sequence of "partial sums" ##s_n = \sum_{i=0}^n a_i##. If this sequence has a limit L then ##\sum_{i=0}^\infty a_i## is
defined to be L. Otherwise, the notation "##a_0 + a_1 + ...##" doesn't represent a number (or a variable).
One may say that the partial sum "##\sum_{i=0}^n a_n## is formed by "truncating the series" as a symbolic expression. But this doesn't assert that we literally "set ##a_{n+1}+a_{n+2}+...## equal to zero". To say we set such a symbolic expression to zero is imprecise because the notation "##a_{n+1}+a_{n+2}+...##" doesn't represent a variable that can be set to zero or any other specific value. For a specific series, the individual symbols "##a_i##" represent constants, not variables. And the notation "##a_{n+1}+a_{n+2}+...##" may not represent any number if that expression, considered as a infinite series in its own right, does not converge.
So the precise way of speaking is merely to say what terms of a series are added up to compute the partial sum ##s_n##, not to assert that the string symbols representing all terms not included in ##s_n## is "set equal to zero".
In browsing online articles about continued fractions, I think it unfortunate that they launch into examples and praise of continued fractions without plainly stating how the notation for continued iterpretations is to be interpreted.
As the mathologer video suggests, the notation for a continued fraction can be interpreted as an infinite series in two different ways. (It's easier to state the partial sums of the series than to state the terms of the series!)
At the moment, I'm having trouble with the forums edit window (,
https://www.physicsforums.com/threa...rs-latex-so-cant-edit-eq.993982/#post-6396002 ) so I'll discuss the possible interpretations in a subsequent post and post this message while the edit window is behaving.
