Infinitely shrinking continued fraction - how to improve design?

Question

To create a continued fraction that truly symbolizes the endless, ever-ongoing nature of an infinite continued fraction, I have taken up this following "notation," where the font size shrinks with every step of the fraction:

I generated it with the following code, compiled with amsmath and graphicx:

$1+\dfrac{1}{2+
\dfrac{1}{\scalebox{.9}{2+{}}
\dfrac{\hfill\scalebox{.9}{1}\hfill}{\scalebox{.8}{2+{}}
\dfrac{\hfill\scalebox{.8}{1}\hfill}{\scalebox{.7}{2+{}}
\dfrac{\hfill\scalebox{.7}{1}\hfill}{\scalebox{.6}{2+{}}
\dfrac{\hfill\scalebox{.6}{1}\hfill}{\scalebox{.5}{2+{}}
\dfrac{\hfill\scalebox{.5}{1}\hfill}{\scalebox{.4}{2+{}}
\dfrac{\hfill\scalebox{.4}{1}\hfill}{\scalebox{.3}{2+{}}
\dfrac{\hfill\scalebox{.3}{1}\hfill}{\scalebox{.2}{2+{}}
}}}}}}}}}$

While I think that the design is decent for now, I would like to know how to fix a few problems in the appearance of the continued fraction, and why these problems arise:

• The spacing of the plus operators get further and further distant from the fraction bar as the descent continues. How can I change my code to make the plus operators look more natural?
• The numerators seem to be drifting away from the fraction bars. How can I refine the spacing between fractions? I suspect this has something to do with \scalebox not changing the size of the box of the fraction's numerator, and only scaling the text, but I haven't figured out how to improve this. Also, if this is the case, why does \scalebox do this?

Also, I feel that my code is in general very inefficient, considering that I need to employ so many braces at the end; solutions that alter the structure of my code are welcome.

Answer 1

It seem to more practical to create a macro which allow you to be more comfortable when you are writing these fractions. For example like this:

\def\fracscale{.8}
\def\etcfrac {\bgroup\def\egroups{\egroup}\etcfracA}
\def\etcfracA#1{\ifx\dots#1\cdots\egroups\else 
   \edef\egroups{\egroup$\egroup 
      \hbox{\pdfsave\pdfsetmatrix{\fracscale\space 0 0 \fracscale}%   
         \dimen0=\fracscale\wd0 \wd0=0pt \ht0=\fracscale\ht0 \dp0=\fracscale\dp0 
         \box0 \pdfrestore \kern\dimen0}\egroups}%
   \setbox0=\hbox\bgroup$\displaystyle\bgroup
   \expandafter\etcfracB\expandafter#1\fi
}
\def\etcfracB#1/#2#3{\dimen0=.8\dimen0
   \strut#1\over #2#3\etcfracA}

\hrule
$  1 + \etcfrac 1/2+1/2+1/2+1/2+1/2+1/2+1/2+\dots $
\hrule

\bye

Answer 2

Avoid applying scaling to text. amsmath has a specific command for continued fractions:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\[
1+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{2+\cfrac{1}
{\dots}}}}}}}}
\]

\end{document}

Answer 3

I wouldn't use this graphic device, but it's not difficult to obtain it:

\documentclass{article}
\usepackage{amsmath,graphicx}

\newcommand{\scaledfrac}[3]{%
  \dfrac{#2}{\scalebox{#1}{$#3$}}%
}

\begin{document}

\[
1+
\scaledfrac{1}{1}{2+
\scaledfrac{.95}{1}{2+
\scaledfrac{.90}{1}{2+
\scaledfrac{.85}{1}{2+
\scaledfrac{.80}{1}{2+
\scaledfrac{.75}{1}{2+
\scaledfrac{.70}{1}{2+
\scaledfrac{.65}{1}{2+
\scaledfrac{.60}{1}{2+
\scaledfrac{.55}{1}{2+\dotsb}
}}}}}}}}}
\]

\end{document}