Actually, the value of $sum after my $sum is undefined.

The following program demonstrates that unless a scalar is set to a value it is initially undefined:

use strict;
my $sum;
unless (defined($sum)) {
    print '$sum is undefined', "\n";
} else {
    print 'The value of $sum is ', "$sum\n";
}
[download]

This program returns the message that $sum is undefined.