Recently while at my local bookstore I found myself browsing through this book. and one of the exercises that caught my eye was to write a program to compute this sum:

   30     29            1
  ---- + ---- + ... + ---- 
    1      2           30
[download]

I immediately thought it would be interesting to compute this exactly as a fraction and also to generalize the problem by replacing 30 with an arbitrary integer n. I knew perl had support for arbitrary precision integers, and I was curious to find out if the numerators or denominators formed any recognizable pattern.

So when I got home I consulted the documentation for bigrat, and naively coded up this loop:

use strict;
use warnings;
use bigrat;

for my $n (1..30) {
  my $s = 0;
  for my $k (1..$n-1) {
    $s += $k / ($n-$k);
  }
  print "n = $n, s = $s\n";
}
[download]

Needless to say, it didn't work as I expected - I got output like this:

n = 1, sum = 0
n = 2, sum = 1
n = 3, sum = 2.5
n = 4, sum = 4.333333333333333
[download]

I expected to get 5/2 for n = 3 and 13/3 for n = 4, etc. After reconsulting the docs and some more experimentation, I finally came up with this solution:

for my $n (1..30) {
  my $s = 0;
  for my $k (1..$n-1) {
    $s += 1/1 * $k / ($n-$k);
  }
  print "n = $n, s = $s\n";
}
[download]

which gave what I wanted:

n = 1, sum = 0
n = 2, sum = 1
n = 3, sum = 5/2
n = 4, sum = 13/3
[download]

Having tackled that problem, I went on to extract the numerator and denominator sequences so I could plug them into The On-Line Encyclopedia of Integer Sequences. Again, my naive approach was to simply call ->numerator and ->denominator on each sum:

for my $n (1..30) {
  ...
  push(@nums, $s->numerator);
  push(@denoms, $s->denominator);
}
[download]

Alas, I ran into problems on the very first term:

Can't locate object method "numerator" via package "Math::BigInt" at .
+/frac line ...
[download]

Indeed, this program illustrates the problem:

use bigrat;
my $x = 1;
print $x->numerator, "\n";
[download]

So I guess my questions are:

1. Why didn't bigrat figure out that the expression $k/($n-$k) should produce a bigrat and not a floating point number?
2. Why can't I call numerator and denominator on a rational that's actually an integer? Should those methods should be added to Math::BigInt or somehow enabled when bigrat is used?

I'm using 5.8.8 if that's of any consequence.

Update 1: fixed typo with expected results.

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