How to calculate successive resultant values of a continued fraction

[swampwiz]

I understand how these expressions are derived, but I don't see how to calculate successive resultant values. Should the continued fraction expression simply be replaced by 1?

 

[jim mcnamara]

Euclidean algorithm? There are limitations regarding irrational results.

https://www.math.u-bordeaux.fr/~pjaming/M1/exposes/MA2.pdf

 

[Stephen Tashi]

I understand how these expressions are derived, but I don't see how to calculate successive resultant values. Should the continued fraction expression simply be replaced by 1?

Notation using "..." can be ambiguous. In the case of continued fractions, there are at least two different ways of associating an infinite sequence with the notation. By tradition, the more interesting way is preferred. For example, begin at 8:55 in the video

 

[swampwiz]

I think folks are missing the gist of my question. Obviously, if one were to use a continued fraction to compute a result, the part of the expression that is still a continuing fraction needs to be given some value just to plug into the arithmetic calculation.

 

[mfb]

Take the integer part of the number: $a_0+x$ where x<1. Take the integer part of 1/x: $a_0+\frac{1}{a_1+y}$ where again y<1. Take the integer part of 1/y and so on.

 

[swampwiz]

The video says to just presume the fraction is 0, and that presuming it to be 1 gives the wrong result.

 

[Stephen Tashi]

The video says to just presume the fraction is 0, and that presuming it to be 1 gives the wrong result.

What fraction are you talking about? A particular fraction? All continued fractions in general?

 

[swampwiz]

What fraction are you talking about? A particular fraction? All continued fractions in general?

All continued fractions in general.

 

[Stephen Tashi]

Your question seems to assume the method for evaluating continued fractions must be done by the "trick" of using a symbol to stand for an infinite expression.

That's the impression I get from your post:

I think folks are missing the gist of my question. Obviously, if one were to use a continued fraction to compute a result, the part of the expression that is still a continuing fraction needs to be given some value just to plug into the arithmetic calculation.

As indicated by @mfb (and the video) the value of a continued fraction is defined as the limit of a sequence, not by whether the trick (also mentioned in the video) works.

It would help if you give a specific example and explain what part of the continued fraction you think must be given a value of 0 or 1. However, evaluating an infinite sequence does not require setting the tail end of the sequence to be 0 or 1. And the definition of how to evaluate a continued fraction does not involve setting the tail of an expression equal to any particular value.

 

[swampwiz]

Your question seems to assume the method for evaluating continued fractions must be done by the "trick" of using a symbol to stand for an infinite expression.

That's the impression I get from your post:
As indicated by @mfb (and the video) the value of a continued fraction is defined as the limit of a sequence, not by whether the trick (also mentioned in the video) works.

It would help if you give a specific example and explain what part of the continued fraction you think must be given a value of 0 or 1. However, evaluating an infinite sequence does not require setting the tail end of the sequence to be 0 or 1. And the definition of how to evaluate a continued fraction does not involve setting the tail of an expression equal to any particular value.

How about for π?

https://en.wikipedia.org/wiki/Pi#Continued_fractions

The Wiki article says:

Truncating the continued fraction at any point yields a rational approximation for π; the first four of these are 3, 22/7, 333/106, and 355/113.

It seems that the first approximation (3) is the result of setting the continuing fraction to 0,

 

[Stephen Tashi]

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.

 

[Stephen Tashi]

Two interpretations of the notation:

$f =\cfrac{b_0}{a_1+\cfrac{b_1}{a_2+\cfrac{b_2}{a_3+\cdots}}}$

Interpretation 1 (mathologer video at 8:16) $f$ is the series whose partial sums are:
$S_0 = \frac{b_0}{a_1}$

$S_1 = \cfrac{ b_0}{a_1 + \frac{b_1}{a_2}}$

$S_2 = \cfrac{b_0} { a_1 + \cfrac{b_1}{ a_2 + \frac{b_2}{a_3} }}$

Interpretation 2: (mathologer video at 8:55) $f$ is the series whose partial sums are:

$S_0 = b_0$

$S_1 = \cfrac{b_0}{a_1 + b_1}$

$S_2 = \cfrac{b_0}{ a_1 + \cfrac{b_1}{ a_2 + b_2} }$

In general, the two interpretations give different numbers. Interpretation 1 is the standard interpretation of the continued fraction $f$.

You may be able to describe the two interpretations informally by talking about setting an infinite string of symbols equal to zero or one. I haven't tried to do that.

 

[swampwiz]

So the standard interpretation is to presume that the continuing fraction is 0.