Problem of the Week 878

One for the New Year

What is the limit of the following monstrous continued fraction as N goes to infinity? Please note that this value is corrected from the original posted version. Thanks to Michael Schweitzer for pointing out the error.

                         1  + ...
                 N  + -----------
                       1999 + ...
       1999 +  ------------------
                       1999 + ...
                 1  + -----------
                         N  + ...
1 + -----------------------------
                       1999 + ...
                 1  +  ----------
                         N  + ...
        N   +  ------------------
                         N  + ...
               1999  + ----------
                         1  + ...

Source: Angel Dorito, Crux. Math Nov 1998


Because of the error, let me spare you the trouble of starting over. It turns out that the answer is the golden ratio. The presence of 1999 is irrelevant. The limit is the golden ratio no matter what real number is used in place of 1999.


Ed Packel and I have just sent out e-fliers and snail mail fliers for our Rocky Mountain Mathematica course this coming July in Colorado. If you would like to receive either, please contact Ed, at:

We teach intro and intermediate courses in Mathematica for one week in Frisco, Colorado (elev. 9100 feet), and have been doing so for six years. July 12-17, 1999.


The second edition of my book, Mathematica in Action, is just out by Springer. It is double the size of the 1990 version, with loads of new stuff: Four Color Theorem, Escher, Banach-Tarski Paradox, Differential Equations, Interpolation, Surface Plotting, and much more. CD-ROM included.

For info contact my editor at Springer, Maria Taylor, at

© Copyright 1999 Stan Wagon. Reproduced with permission.

9 February 1999